We show that the Euler-Mascheroni constant <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M3" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 6.63704 9.39034" width="6.63704pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-225"/></g></svg> and Euler’s number <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M4" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 5.50181 6.1673" width="5.50181pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-102"/></g></svg> can both be represented as a product of a Riordan matrix and certain row and column vectors. 

Edray Herber Goins

Asamoah Nkwanta

Riordan Matrix Representations of Euler&#x2019;s Constant <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42943pt" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 6.63704 9.39034" width="6.63704pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-225" d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" /></g></svg> and Euler&#x2019;s Number <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.2063904pt" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 5.50181 6.1673" width="5.50181pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-102" d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" /></g></svg>

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M1" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g></svg> be a group and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M2" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 6.25863 8.8423" width="6.25863pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g></svg> a nonempty subset of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M3" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-72"/></g></svg>. Then, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 6.25863 8.8423" width="6.25863pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-84"/></g></svg> is<i> product-free</i> if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.45907pt" id="M5" height="10.7474pt" version="1.1" viewBox="-0.0498162 -9.28833 33.1346 10.7474" width="33.1346pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.013,0,0,-0.013,6.58,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g><g transform="matrix(.013,0,0,-0.013,16.313,0)"><path id="g117-181" d="M448 1V51H364C325 51 289 60 257 75L304 230H448V280H320L374 459H448V509H389L420 610L366 627L329 507C189 491 80 384 80 255C80 173 123 101 191 54L144 -99L199 -116L242 26C279 10 321 1 365 1H448ZM245 230L207 108C171 139 146 182 140 230H245ZM313 454L260 280H140C151 366 221 435 313 454Z"/></g><g transform="matrix(.013,0,0,-0.013,26.809,0)"><use xlink:href="#g113-84"/></g></svg> for all <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.57657pt" id="M6" height="10.8649pt" version="1.1" viewBox="-0.0498162 -9.28833 38.2888 10.8649" width="38.2888pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.013,0,0,-0.013,6.58,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.013,0,0,-0.013,11.723,0)"><use xlink:href="#g113-99"/></g><g transform="matrix(.013,0,0,-0.013,21.456,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.013,0,0,-0.013,31.952,0)"><use xlink:href="#g113-84"/></g></svg>. We say <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M7" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 6.25863 8.8423" width="6.25863pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-84"/></g></svg> is a<i> locally maximal product-free set</i> if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M8" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 6.25863 8.8423" width="6.25863pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-84"/></g></svg> is product-free and not properly contained in any other product-free set. It is natural to ask whether it is possible to determine the smallest possible size of a locally maximal product-free set in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M9" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-72"/></g></svg>. Alternatively, given a positive integer <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M10" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>, one can ask the following: what is the largest integer <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M11" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 11.7322 9.25202" width="11.7322pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.461,3.132)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"/></g></svg> such that there is a group of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M12" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 11.7322 9.25202" width="11.7322pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0091,0,0,-0.0091,6.461,3.132)"><use xlink:href="#g50-108"/></g></svg> with a locally maximal product-free set of size <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M13" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>? The groups containing locally maximal product-free sets of sizes <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M14" height="8.39864pt" version="1.1" viewBox="-0.0498162 -8.34882 6.36303 8.39864" width="6.36303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M15" height="8.39864pt" version="1.1" viewBox="-0.0498162 -8.34882 6.36303 8.39864" width="6.36303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg> are known, and it has been conjectured that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29108pt" id="M16" height="11.6399pt" version="1.1" viewBox="-0.0498162 -8.34882 38.9814 11.6399" width="38.9814pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0091,0,0,-0.0091,6.461,3.132)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g><g transform="matrix(.013,0,0,-0.013,15.039,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,26.302,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,32.542,0)"><path id="g113-53" d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"/></g></svg>. The purpose of this paper is to prove this conjecture and hence show that the list of known locally maximal product-free sets of size 3 is complete. We also report some experimental observations about the sequence <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M17" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 11.7322 9.25202" width="11.7322pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0091,0,0,-0.0091,6.461,3.132)"><use xlink:href="#g50-108"/></g></svg>.

Chimere S. Anabanti

Sarah B. Hart

Groups Containing Small Locally Maximal Product-Free Sets

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M1" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 6.50656 9.39034" width="6.50656pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg> series identities.

S. M. Ripon

A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

We define a two-player combinatorial game in which players take alternate turns; each turn consists of deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player&#x2019;s move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph&#x2019;s automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.

Richard Adams

Janae Dixon

Jennifer Elder

Jamie Peabody

Oscar Vega

Karen Willis

Combinatorial Analysis of a Subtraction Game on Graphs

A graph <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M1" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g></svg> is said to be a self-centered graph if the eccentricity of every vertex of the graph is the same. In other words, a graph is a self-centered graph if radius and diameter of the graph are equal. In this paper, self-centeredness of strong product, co-normal product, and lexicographic product of graphs is studied in detail. The necessary and sufficient conditions for these products of graphs to be a self-centered graph are also discussed. The distance between any two vertices in the co-normal product of a finite number of graphs is also computed analytically.

Priyanka Singh

Pratima Panigrahi

On Self-Centeredness of Product of Graphs

We propose a procedure of constructing new block designs starting from a given one by looking at the intersections of its blocks with various sets and grouping those sets according to the structure of the intersections. We introduce a symmetric relationship of friendship between block designs built on a set <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M1" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.35121 8.8423" width="9.35121pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-87" d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"/></g></svg> and consider families of block designs where all designs are friends of each other, the so-called friendly families. We show that a friendly family admits a partial ordering. Furthermore, we exhibit a map from the power set of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M2" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.35121 8.8423" width="9.35121pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-87"/></g></svg>, partially ordered by inclusion, to a friendly family of a particular type which preserves the partial order.

Alexander Shramchenko

Vasilisa Shramchenko

A New Relationship between Block Designs

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M1" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg> cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M2" height="8.55521pt" version="1.1" viewBox="-0.0498162 -8.34882 12.9135 8.55521" width="12.9135pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-53" d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg> and weight <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M3" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M4" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g></svg> is an odd positive integer. Then we show that, for any square <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M5" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>, there is an integer <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M6" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 26.6408 11.5564" width="26.6408pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g><g transform="matrix(.013,0,0,-0.013,10.91,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,15.408,0)"><use xlink:href="#g113-108"/></g><g transform="matrix(.013,0,0,-0.013,21.947,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> such that, for each <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M7" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 48.1146 11.5564" width="48.1146pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.013,0,0,-0.013,10.158,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,21.421,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,32.331,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,36.829,0)"><use xlink:href="#g113-108"/></g><g transform="matrix(.013,0,0,-0.013,43.368,0)"><use xlink:href="#g113-42"/></g></svg>, there is a symmetric weighing matrix of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M8" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g></svg> and weight <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M9" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.

Ebrahim Ghaderpour

Some Nonexistence and Asymptotic Existence Results for Weighing Matrices

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International Journal of Combinatorics