A subset <svg style="vertical-align:-0.0pt;width:12.0375px;" id="M1" height="9.2749996" version="1.1" viewBox="0 0 12.0375 9.2749996" width="12.0375"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,9.275)">
		<g transform="translate(72,-64.58)">
			<text transform="matrix(1,0,0,-1,-71.95,64.63)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑋</tspan>
			</text>

		</g>
	</g>
</svg> in the <svg style="vertical-align:-0.12375pt;width:7.7125001px;" id="M2" height="9.6374998" version="1.1" viewBox="0 0 7.7125001 9.6374998" width="7.7125001"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,9.6375)">
		<g transform="translate(72,-64.29)">
			<text transform="matrix(1,0,0,-1,-71.95,64.46)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑘</tspan>
			</text>

		</g>
	</g>
</svg>-dimensional Euclidean space <svg style="vertical-align:-0.0pt;width:16.525px;" id="M3" height="12.275" version="1.1" viewBox="0 0 16.525 12.275" width="16.525"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.275)">
		<g transform="translate(72,-62.18)">
			<text transform="matrix(1,0,0,-1,-71.95,62.23)">
				<tspan style="font-size: 11.21px; " x="0" y="0">ℝ</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-63.58,66.71)">
				<tspan style="font-size: 7.85px; " x="0" y="0">𝑘</tspan>
			</text>

		</g>
	</g>
</svg> that contains <svg style="vertical-align:-0.09pt;width:7.0875001px;" id="M4" height="6.4124999" version="1.1" viewBox="0 0 7.0875001 6.4124999" width="7.0875001"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,6.4125)">
		<g transform="translate(72,-66.87)">
			<text transform="matrix(1,0,0,-1,-71.95,67.01)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg> points (elements) is
called an <svg style="vertical-align:-0.09pt;width:7.0875001px;" id="M5" height="6.4124999" version="1.1" viewBox="0 0 7.0875001 6.4124999" width="7.0875001"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,6.4125)">
		<g transform="translate(72,-66.87)">
			<text transform="matrix(1,0,0,-1,-71.95,67.01)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg>-point isosceles set if every triplet of points selected from them forms an isosceles
triangle. In this paper, we show that there exist exactly two 11-point isosceles sets in <svg style="vertical-align:-0.0pt;width:16.112499px;" id="M6" height="12.3625" version="1.1" viewBox="0 0 16.112499 12.3625" width="16.112499"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.3625)">
		<g transform="translate(72,-62.11)">
			<text transform="matrix(1,0,0,-1,-71.95,62.16)">
				<tspan style="font-size: 11.21px; " x="0" y="0">ℝ</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-63.58,66.65)">
				<tspan style="font-size: 7.85px; " x="0" y="0">4</tspan>
			</text>

		</g>
	</g>
</svg> up to
isomorphisms and that the maximum cardinality of isosceles sets in <svg style="vertical-align:-0.0pt;width:16.112499px;" id="M7" height="12.3625" version="1.1" viewBox="0 0 16.112499 12.3625" width="16.112499"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.3625)">
		<g transform="translate(72,-62.11)">
			<text transform="matrix(1,0,0,-1,-71.95,62.16)">
				<tspan style="font-size: 11.21px; " x="0" y="0">ℝ</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-63.58,66.65)">
				<tspan style="font-size: 7.85px; " x="0" y="0">4</tspan>
			</text>

		</g>
	</g>
</svg> is 11.

Hiroaki Kido

On Isosceles Sets in the 4-Dimensional Euclidean Space

Dragomir &#x17D;. &#x110;okovi&#x107;

Erratum to &#x201C;Classification of Base Sequences <svg style="vertical-align:-2.3205pt" height="15.0875" version="1.1" viewBox="0 0 79.412498 15.0875" width="79.412498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x42" d="M372 352v-2q176 -27 176 -159q0 -99 -95 -151q-73 -40 -196 -40h-221v28q65 5 78.5 19t13.5 76v404q0 63 -12 77t-72 18v28h257q102 0 150 -34q26 -18 41 -48.5t15 -65.5q0 -111 -135 -150zM213 362h47q80 0 117.5 32t37.5 96q0 56 -33 92t-100 36q-41 0 -56 -10&#xA;q-13 -8 -13 -50v-196zM213 328v-203q0 -54 17 -72.5t66 -18.5q67 1 110.5 37t43.5 110q0 147 -192 147h-45z" /></g><g transform="matrix(.017,-0,0,-.017,10.058,12.138)"><path id="x53" d="M409 504l-29 -5q-16 60 -44.5 96t-86.5 36q-54 0 -82.5 -32t-28.5 -77q0 -51 30.5 -82.5t96.5 -65.5l35.5 -18t33 -19.5t33.5 -23.5l27 -25.5t24.5 -32t13.5 -36.5t6 -45q0 -80 -62 -134.5t-160 -54.5q-47 0 -98 15q-22 7 -50 21q-8 23 -27 155l30 7q7 -27 18 -52&#xA;t30 -51.5t49 -42.5t67 -16q56 0 88 32.5t32 87.5q0 51 -31.5 82t-98.5 67q-80 44 -110 73q-55 53 -55 124q0 75 56 126.5t150 51.5q53 0 126 -23z" /></g><g transform="matrix(.017,-0,0,-.017,18.115,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,23.997,12.138)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,36.304,12.138)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,50.056,12.138)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,58.215,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,64.93,12.138)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,73.463,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg>&#x201D;

We discuss some problems and permutation statistics involving two different types of random permutations. Under the usual model of random permutations, we prove that the shifted coverage of the elements of <svg style="vertical-align:-2.26974pt;width:15.4375px;" id="M1" height="13.575" version="1.1" viewBox="0 0 15.4375 13.575" width="15.4375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.575)">
		<g transform="translate(72,-61.14)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">{</tspan>
				<tspan style="font-size: 12.50px; " x="6.00144" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg>, 2, <svg style="vertical-align:-0.13794pt;width:18.362499px;" id="M2" height="1.8625" version="1.1" viewBox="0 0 18.362499 1.8625" width="18.362499"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,1.8625)">
		<g transform="translate(72,-70.51)">
			<text transform="matrix(1,0,0,-1,-71.95,70.7)">
				<tspan style="font-size: 12.50px; " x="0" y="0">…</tspan>
			</text>

		</g>
	</g>
</svg>, <svg style="vertical-align:-2.26974pt;width:16.1px;" id="M3" height="13.575" version="1.1" viewBox="0 0 16.1 13.575" width="16.1"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.575)">
		<g transform="translate(72,-61.14)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑘</tspan>
				<tspan style="font-size: 12.50px; " x="6.7766261" y="0">}</tspan>
			</text>

		</g>
	</g>
</svg> of a random permutation over <svg style="vertical-align:-2.26974pt;width:15.4375px;" id="M4" height="13.575" version="1.1" viewBox="0 0 15.4375 13.575" width="15.4375"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.575)">
		<g transform="translate(72,-61.14)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">{</tspan>
				<tspan style="font-size: 12.50px; " x="6.00144" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg>, 2, <svg style="vertical-align:-0.13794pt;width:18.362499px;" id="M5" height="1.8625" version="1.1" viewBox="0 0 18.362499 1.8625" width="18.362499"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,1.8625)">
		<g transform="translate(72,-70.51)">
			<text transform="matrix(1,0,0,-1,-71.95,70.7)">
				<tspan style="font-size: 12.50px; " x="0" y="0">…</tspan>
			</text>

		</g>
	</g>
</svg>, <svg style="vertical-align:-2.26974pt;width:15.4px;" id="M6" height="13.575" version="1.1" viewBox="0 0 15.4 13.575" width="15.4"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.575)">
		<g transform="translate(72,-61.14)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="6.2139912" y="0">}</tspan>
			</text>

		</g>
	</g>
</svg>; that is, the size of the union of the cycles containing these elements, excluding these elements themselves, follows a negative hypergeometric distribution. This fact gives a probabilistic model for the coverage via the canonical cycle representation. For a different random model, we determine some random permutation statistics regarding the problem of the
lost boarding pass and its variations.

Tam&#225;s Lengyel

The Distribution of the Size of the Union of Cycles for Two Types of  Random Permutations

The 157, 211 triangle-free symmetric <svg style="vertical-align:-3.3907pt;width:22.775px;" id="M2" height="15.1375" version="1.1" viewBox="0 0 22.775 15.1375" width="22.775" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x32" /></g><g transform="matrix(.017,-0,0,-.017,8.222,10.862)"><use xlink:href="#x32" /></g><g transform="matrix(.012,-0,0,-.012,16.375,14.938)"><use xlink:href="#x33" /></g></svg> configurations are classified and some of their
properties are examined. We conclude that each such configuration has a blocking set. Further
properties like transitivity on lines, self-duality, and self-polarity are discussed.

Abdullah Al-Azemi

Anton Betten

Classification of Triangle-Free <svg style="vertical-align:-3.39069pt" height="15.1375" version="1.1" viewBox="0 0 22.775 15.1375" width="22.775" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g><g transform="matrix(.017,-0,0,-.017,8.222,10.862)"><use xlink:href="#x32"/></g><g transform="matrix(.012,-0,0,-.012,16.375,14.938)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g></svg> Configurations

It is known that every class-regular symmetric <svg style="vertical-align:-2.29482pt;width:37.662498px;" id="M1" height="13.55" version="1.1" viewBox="0 0 37.662498 13.55" width="37.662498"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.55)">
		<g transform="translate(72,-61.16)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜇</tspan>
				<tspan style="font-size: 12.50px; " x="11.765323" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="16.979074" y="0">𝑚</tspan>
				<tspan style="font-size: 12.50px; " x="25.856203" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>-net is tactical. Also it is known that all <svg style="vertical-align:-2.29482pt;width:37.662498px;" id="M2" height="13.55" version="1.1" viewBox="0 0 37.662498 13.55" width="37.662498"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.55)">
		<g transform="translate(72,-61.16)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜇</tspan>
				<tspan style="font-size: 12.50px; " x="11.765323" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="16.979074" y="0">𝑚</tspan>
				<tspan style="font-size: 12.50px; " x="25.856203" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>-nets with <svg style="vertical-align:-0.10033pt;width:38.424999px;" id="M3" height="10.8125" version="1.1" viewBox="0 0 38.424999 10.8125" width="38.424999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.8125)">
		<g transform="translate(72,-63.35)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
				<tspan style="font-size: 12.50px; " x="12.340461" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="24.38085" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg> or <svg style="vertical-align:-2.29482pt;width:36.849998px;" id="M4" height="13.55" version="1.1" viewBox="0 0 36.849998 13.55" width="36.849998"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.55)">
		<g transform="translate(72,-61.16)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
				<tspan style="font-size: 12.50px; " x="11.077658" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="23.118048" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg> are tactical. In the work of Al-Kenani and Mavron (2003), it is proved that every symmetric net with <svg style="vertical-align:-0.17555pt;width:38.424999px;" id="M5" height="10.9125" version="1.1" viewBox="0 0 38.424999 10.9125" width="38.424999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
		<g transform="translate(72,-63.27)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
				<tspan style="font-size: 12.50px; " x="12.340461" y="0">=</tspan>
				<tspan style="font-size: 12.50px; " x="24.38085" y="0">3</tspan>
			</text>

		</g>
	</g>
</svg> is tactical if and only if it is class regular. In this paper, we construct <svg style="vertical-align:-2.21957pt;width:32.674999px;" id="M6" height="13.45" version="1.1" viewBox="0 0 32.674999 13.45" width="32.674999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">2</tspan>
				<tspan style="font-size: 12.50px; " x="10.414999" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="15.616247" y="0">4</tspan>
				<tspan style="font-size: 12.50px; " x="21.867746" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>-net and show that it
is class regular and therefore tactical. New necessary and sufficient conditions are given for a symmetric net to admit a nonidentity bitranslation.

Ahmad N. Al-Kenani

Bitranslations and Symmetric Nets

Base sequences BS<svg style="vertical-align:-2.21957pt;width:58.049999px;" id="M2" height="13.45" version="1.1" viewBox="0 0 58.049999 13.45" width="58.049999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="13.153156" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="24.493378" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="30.744877" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="35.95863" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="42.172619" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>
 are quadruples of <svg style="vertical-align:-2.26974pt;width:33.650002px;" id="M3" height="13.575" version="1.1" viewBox="0 0 33.650002 13.575" width="33.650002"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.575)">
		<g transform="translate(72,-61.14)">
			<text transform="matrix(1,0,0,-1,-71.95,63.45)">
				<tspan style="font-size: 12.50px; " x="0" y="0">{</tspan>
				<tspan style="font-size: 12.50px; " x="6.00144" y="0">±</tspan>
				<tspan style="font-size: 12.50px; " x="14.565995" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="20.817495" y="0">}</tspan>
			</text>

		</g>
	</g>
</svg>-sequences <svg style="vertical-align:-2.21957pt;width:76.300003px;" id="M4" height="13.45" version="1.1" viewBox="0 0 76.300003 13.45" width="76.300003"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝐴</tspan>
				<tspan style="font-size: 12.50px; " x="13.12815" y="0">;</tspan>
				<tspan style="font-size: 12.50px; " x="18.679482" y="0">𝐵</tspan>
				<tspan style="font-size: 12.50px; " x="27.381571" y="0">;</tspan>
				<tspan style="font-size: 12.50px; " x="32.945404" y="0">𝐶</tspan>
				<tspan style="font-size: 12.50px; " x="41.334919" y="0">;</tspan>
				<tspan style="font-size: 12.50px; " x="46.88625" y="0">𝐷</tspan>
				<tspan style="font-size: 12.50px; " x="56.763618" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>, with <i  >A</i> and <i  >B</i> of length <svg style="vertical-align:-0.51414pt;width:33.362499px;" id="M5" height="11.325" version="1.1" viewBox="0 0 33.362499 11.325" width="33.362499"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,11.325)">
		<g transform="translate(72,-62.94)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="8.9896574" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="20.329878" y="0">1</tspan>
			</text>

		</g>
	</g>
</svg> and <i  >C</i> and <i  >D</i> of length <i  >n</i>, such that the sum of their nonperiodic autocor-relation functions is a <svg style="vertical-align:-0.13794pt;width:7.9875002px;" id="M6" height="10.7375" version="1.1" viewBox="0 0 7.9875002 10.7375" width="7.9875002"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.7375)">
		<g transform="translate(72,-63.41)">
			<text transform="matrix(1,0,0,-1,-71.95,63.6)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝛿</tspan>
			</text>

		</g>
	</g>
</svg>-function. The base sequence conjecture, asserting that BS<svg style="vertical-align:-2.21957pt;width:58.049999px;" id="M7" height="13.45" version="1.1" viewBox="0 0 58.049999 13.45" width="58.049999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="13.153156" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="24.493378" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="30.744877" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="35.95863" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="42.172619" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>
 exist for all <i  >n</i>, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS<svg style="vertical-align:-2.21957pt;width:58.049999px;" id="M8" height="13.45" version="1.1" viewBox="0 0 58.049999 13.45" width="58.049999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="13.153156" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="24.493378" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="30.744877" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="35.95863" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="42.172619" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS<svg style="vertical-align:-2.21957pt;width:58.049999px;" id="M9" height="13.45" version="1.1" viewBox="0 0 58.049999 13.45" width="58.049999"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
		<g transform="translate(72,-61.24)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="13.153156" y="0">+</tspan>
				<tspan style="font-size: 12.50px; " x="24.493378" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="30.744877" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="35.95863" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="42.172619" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>
 for <svg style="vertical-align:-1.29163pt;width:42.912498px;" id="M10" height="12.3" version="1.1" viewBox="0 0 42.912498 12.3" width="42.912498"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,12.3)">
		<g transform="translate(72,-62.16)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="9.6773224" y="0">≤</tspan>
				<tspan style="font-size: 12.50px; " x="21.71771" y="0">3</tspan>
				<tspan style="font-size: 12.50px; " x="27.969212" y="0">0</tspan>
			</text>

		</g>
	</g>
</svg>. As the number of equivalence classes grows rapidly (but not monotonically) with <i  >n</i>, the tables in the paper cover only the cases <svg style="vertical-align:-1.29163pt;width:42.912498px;" id="M11" height="12.3" version="1.1" viewBox="0 0 42.912498 12.3" width="42.912498"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,12.3)">
		<g transform="translate(72,-62.16)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
				<tspan style="font-size: 12.50px; " x="9.6773224" y="0">≤</tspan>
				<tspan style="font-size: 12.50px; " x="21.71771" y="0">1</tspan>
				<tspan style="font-size: 12.50px; " x="27.969212" y="0">3</tspan>
			</text>

		</g>
	</g>
</svg>.

Dragomir &#381;. &#208;okovi&#263;

Classification of Base Sequences <svg style="vertical-align:-2.3205pt" height="15.0875" version="1.1" viewBox="0 0 80 15.0875" width="80" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g><g transform="matrix(.017,-0,0,-.017,10.551,12.138)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g><g transform="matrix(.017,-0,0,-.017,18.71,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,24.592,12.138)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,36.899,12.138)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,50.651,12.138)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,58.81,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,65.525,12.138)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,74.058,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> 

A procedure for the construction of Costas arrays of infinite size representing configurations on non-attacking
queens on the chessboard is presented.

Konstantinos Drakakis

On the Existence of Infinite Size Costas Arrays Configurations of Nonattacking Queens on the Chessboard

The existence of a class regular symmetric transversal design <svg style="vertical-align:-3.3907pt;width:82.324997px;" id="M1" height="16.450001" version="1.1" viewBox="0 0 82.324997 16.450001" width="82.324997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x53" d="M409 504l-29 -5q-16 60 -44.5 96t-86.5 36q-54 0 -82.5 -32t-28.5 -77q0 -51 30.5 -82.5t96.5 -65.5l35.5 -18t33 -19.5t33.5 -23.5l27 -25.5t24.5 -32t13.5 -36.5t6 -45q0 -80 -62 -134.5t-160 -54.5q-47 0 -98 15q-22 7 -50 21q-8 23 -27 155l30 7q7 -27 18 -52&#xA;t30 -51.5t49 -42.5t67 -16q56 0 88 32.5t32 87.5q0 51 -31.5 82t-98.5 67q-80 44 -110 73q-55 53 -55 124q0 75 56 126.5t150 51.5q53 0 126 -23z" /></g><g transform="matrix(.017,-0,0,-.017,7.95,12.162)"><path id="x54" d="M592 498l-30 -3q-13 63 -31 89q-13 19 -33.5 25.5t-75.5 6.5h-70v-493q0 -60 16 -75.5t86 -19.5v-28h-286v28q68 4 83.5 19.5t15.5 75.5v493h-62q-60 0 -83 -7t-33 -24q-12 -17 -32 -90h-29q10 116 12 180h20q10 -16 21 -20.5t34 -4.5h395q19 0 28.5 5t21.5 20h21&#xA;q2 -89 11 -177z" /></g><g transform="matrix(.017,-0,0,-.017,18.438,12.162)"><path id="x44" d="M43 650h280q194 0 282 -85q85 -82 85 -215q0 -173 -123 -267q-107 -83 -297 -83h-236v27q65 5 79.5 19.5t14.5 77.5v402q0 45 -6 62.5t-21.5 24t-57.5 9.5v28zM213 547v-416q0 -55 24.5 -75t81.5 -20q133 0 203 86t70 214q0 80 -30 143t-88 97q-66 41 -170 41&#xA;q-62 0 -78 -13q-13 -9 -13 -57z" /></g><g transform="matrix(.012,-0,0,-.012,30.938,16.25)"><path id="x1D706" d="M529 97q-70 -109 -136 -109q-41 0 -56 94q-23 144 -37 284q-38 -88 -99 -202.5t-93 -156.5q-26 -8 -76 -19l-9 21q71 78 145.5 193t124.5 232q-5 84 -15 128q-12 55 -29.5 75.5t-42.5 20.5q-21 0 -45 -13l-8 24q16 17 46 30t55 13q43 0 70 -46.5t40 -169.5&#xA;q27 -249 51 -392q7 -46 23 -46q24 0 70 60z" /></g><g transform="matrix(.017,-0,0,-.017,38.125,12.162)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,43.99,12.162)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.017,-0,0,-.017,52.149,12.162)"><use xlink:href="#x1D706" /></g><g transform="matrix(.017,-0,0,-.017,61.532,12.162)"><path id="x3B" d="M114 412q23 0 39 -16t16 -41t-16 -41.5t-40 -16.5t-39.5 16.5t-15.5 41.5t15.5 41t40.5 16zM95 130q31 0 61 -30t30 -78q0 -53 -38 -88t-92 -52l-11 30q76 32 76 85q0 26 -17.5 43.5t-44.5 23.5q-13 3 -13 24q0 19 14.5 30.5t34.5 11.5z" /></g><g transform="matrix(.017,-0,0,-.017,68.23,12.162)"><use xlink:href="#x33" /></g><g transform="matrix(.017,-0,0,-.017,76.389,12.162)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g></svg> is equivalent to a generalized Hadamard matrix of order <svg style="vertical-align:-0.1638pt;width:17.424999px;" id="M2" height="11.125" version="1.1" viewBox="0 0 17.424999 11.125" width="17.424999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x33" /></g><g transform="matrix(.017,-0,0,-.017,8.222,10.862)"><path id="x1D462" d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74&#xA;t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" /></g></svg> over <svg style="vertical-align:-2.3205pt;width:41.1875px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 41.1875 15.0875" width="41.1875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x47" d="M692 302v-28q-45 -6 -54.5 -17t-9.5 -55v-86q0 -62 7 -95q-7 -2 -32 -7.5l-37 -8l-35.5 -7.5t-40.5 -7t-38.5 -4t-40.5 -2q-165 0 -266 93t-101 243q0 84 32 151t86 108t121.5 63t142.5 22q29 0 62.5 -4t52 -7.5t47.5 -10.5t31 -7l15 -156l-28 -6q-17 87 -66.5 121.5&#xA;t-128.5 34.5q-117 0 -191 -80.5t-74 -216.5q0 -56 17 -110t49.5 -100.5t87.5 -74.5t124 -28q53 0 87 17q17 9 24.5 23t7.5 43v85q0 54 -15.5 67.5t-89.5 18.5v28h254z" /></g><g transform="matrix(.017,-0,0,-.017,12.217,12.138)"><path id="x46" d="M493 503l-29 -4q-13 56 -28 80q-13 22 -35.5 29.5t-78.5 7.5h-76q-22 0 -27.5 -5t-5.5 -27v-238h103q66 0 82.5 12t25.5 62h29v-193h-29q-9 54 -25.5 67t-82.5 13h-103v-181q0 -64 15 -78.5t84 -19.5v-28h-272v28q61 5 74.5 19.5t13.5 78.5v399q0 64 -13 78t-72 19v28&#xA;h440q1 -40 10 -147z" /></g><g transform="matrix(.017,-0,0,-.017,21.209,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,27.091,12.138)"><use xlink:href="#x33" /></g><g transform="matrix(.017,-0,0,-.017,35.25,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg>. Let <svg style="vertical-align:-3.3907pt;width:15.85px;" id="M4" height="11.9625" version="1.1" viewBox="0 0 15.85 11.9625" width="15.85" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.012,-0,0,-.012,8.6,11.762)"><use xlink:href="#x1D706" /></g></svg> be the number of nonisomorphic <svg style="vertical-align:-3.3907pt;width:82.324997px;" id="M5" height="16.450001" version="1.1" viewBox="0 0 82.324997 16.450001" width="82.324997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x53" /></g><g transform="matrix(.017,-0,0,-.017,7.95,12.162)"><use xlink:href="#x54" /></g><g transform="matrix(.017,-0,0,-.017,18.438,12.162)"><use xlink:href="#x44" /></g><g transform="matrix(.012,-0,0,-.012,30.938,16.25)"><use xlink:href="#x1D706" /></g><g transform="matrix(.017,-0,0,-.017,38.125,12.162)"><use xlink:href="#x5B" /></g><g transform="matrix(.017,-0,0,-.017,43.99,12.162)"><use xlink:href="#x33" /></g><g transform="matrix(.017,-0,0,-.017,52.149,12.162)"><use xlink:href="#x1D706" /></g><g transform="matrix(.017,-0,0,-.017,61.532,12.162)"><use xlink:href="#x3B" /></g><g transform="matrix(.017,-0,0,-.017,68.23,12.162)"><use xlink:href="#x33" /></g><g transform="matrix(.017,-0,0,-.017,76.389,12.162)"><use xlink:href="#x5D" /></g></svg>'s. It is known that <svg style="vertical-align:-3.27605pt;width:42.575001px;" id="M6" height="15" version="1.1" viewBox="0 0 42.575001 15" width="42.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.938)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.862)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.862)"><use xlink:href="#x31" /></g></svg>, <svg style="vertical-align:-3.27605pt;width:42.575001px;" id="M7" height="15" version="1.1" viewBox="0 0 42.575001 15" width="42.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.938)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.862)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.862)"><use xlink:href="#x31" /></g></svg>, <svg style="vertical-align:-3.3907pt;width:42.575001px;" id="M8" height="15.0875" version="1.1" viewBox="0 0 42.575001 15.0875" width="42.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.8)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.887)"><use xlink:href="#x33" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.8)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.8)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g></svg>, <svg style="vertical-align:-3.27605pt;width:42.575001px;" id="M9" height="15" version="1.1" viewBox="0 0 42.575001 15" width="42.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.938)"><use xlink:href="#x34" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.862)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.862)"><use xlink:href="#x31" /></g></svg>, <svg style="vertical-align:-3.3907pt;width:42.575001px;" id="M10" height="15.1375" version="1.1" viewBox="0 0 42.575001 15.1375" width="42.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.938)"><path id="x35" d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8&#xA;l-26 -65h-233z" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.862)"><use xlink:href="#x3D" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.862)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g></svg>, <svg style="vertical-align:-3.3907pt;width:50.737499px;" id="M11" height="15.1375" version="1.1" viewBox="0 0 50.737499 15.1375" width="50.737499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.938)"><path id="x36" d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5&#xA;t96 -62.5q53 0 78 45.5t25 105.5q0 68 -35 116t-99 48z" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.862)"><path id="x2265" d="M531 285l-474 -214v56l416 183l-416 184v56l474 -215v-50zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.862)"><use xlink:href="#x32" /></g><g transform="matrix(.017,-0,0,-.017,42.496,10.862)"><use xlink:href="#x30" /></g></svg>, and <svg style="vertical-align:-3.3907pt;width:42.575001px;" id="M12" height="14.9375" version="1.1" viewBox="0 0 42.575001 14.9375" width="42.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.65)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.737)"><path id="x37" d="M447 623l8 -12l-283 -613l-74 -10l-7 11q174 283 297 551h-216q-48 0 -62.5 -12t-33.5 -63h-29q10 60 18 148h382z" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.65)"><use xlink:href="#x2265" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.65)"><use xlink:href="#x35" /></g></svg>. In this paper, it is shown that <svg style="vertical-align:-3.3907pt;width:50.737499px;" id="M13" height="15.1375" version="1.1" viewBox="0 0 50.737499 15.1375" width="50.737499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B" /></g><g transform="matrix(.012,-0,0,-.012,8.6,14.938)"><path id="x38" d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5&#xA;t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" /></g><g transform="matrix(.017,-0,0,-.017,19.65,10.862)"><use xlink:href="#x2265" /></g><g transform="matrix(.017,-0,0,-.017,34.337,10.862)"><use xlink:href="#x32" /></g><g transform="matrix(.017,-0,0,-.017,42.496,10.862)"><use xlink:href="#x34" /></g></svg>.

Teppei Hatono

Chihiro Suetake

On Symmetric Transversal Designs STD<sub><b>8</b></sub>[24; 3]'s

It was shown  by Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from Kirschenhofer and Prodinger (1998) and Kuba et al. (2008)  can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.

Markus Kuba

Helmut Prodinger

On a Reciprocity Law for Finite Multiple Zeta Values

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International Journal of Combinatorics