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Discrete Dynamics in Nature and Society
VolumeΒ 2012Β (2012), Article IDΒ 760246, 17 pages
http://dx.doi.org/10.1155/2012/760246
Research Article

The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group

1Department of Mathematics, Kangnung-Wonju National University, Kangnung 210-702, Republic of Korea
2Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 9 May 2012; Accepted 25 July 2012

Academic Editor: Prasanta K.Β Panigrahi

Copyright Β© 2012 Ju-Mok Oh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dicyclic group 𝐡4𝑛 of order 4𝑛 by finding its generating function of multivariables.

1. Introduction

Throughout this paper, all groups are assumed to be finite. The lattice of subgroups of a given group 𝐺 is the lattice (𝐿(𝐺),≀) where 𝐿(𝐺) is the set of all subgroups of 𝐺 and the partial order ≀ is the set inclusion. In this lattice (𝐿(𝐺),≀), a chain of subgroups of 𝐺 is a subset of 𝐿(𝐺) linearly ordered by set inclusion. A chain of subgroups of 𝐺 is called 𝐺-rooted (or rooted) if it contains 𝐺. Otherwise, it is called unrooted.

The problem of counting chains of subgroups of a given group 𝐺 has received attention by researchers with related to classifying fuzzy subgroups of 𝐺 under a certain type of equivalence relation. Some works have been done on the particular families of finite abelian groups (e.g., see [1–4]). As a step of this problem toward non-abelian groups, the first author [5] has found an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dihedral group 𝐷2𝑛 of order 2𝑛 where 𝑛 is an arbitrary positive integer. As a continuation of this work, we give an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dicyclic group 𝐡4𝑛 of order 4𝑛 by finding its generating function of multivariables where 𝑛 is an arbitrary integer.

2. Preliminaries

Given a group 𝐺, let π’ž(𝐺), 𝒰(𝐺), and β„›(𝐺) be the collection of chains of subgroups of 𝐺, of unrooted chains of subgroups of 𝐺, and of 𝐺-rooted chains of subgroups of 𝐺, respectively. Let 𝐢(𝐺)∢=|π’ž(𝐺)|, π‘ˆ(𝐺)∢=|𝒰(𝐺)|, and 𝑅(𝐺)∢=|β„›(𝐺)|.

The following simple observation is useful for enumerating chains of subgroups of a given group.

Proposition 2.1. Let 𝐺 be a finite group. Then 𝑅(𝐺)=π‘ˆ(𝐺)+1 and 𝐢(𝐺)=𝑅(𝐺)+π‘ˆ(𝐺)=2𝑅(𝐺)βˆ’1.

For a fixed positive integer π‘˜, we define a function πœ† as follows: πœ†ξ€·π‘₯π‘˜ξ€ΈβˆΆ=1βˆ’2π‘₯π‘˜,πœ†ξ€·π‘₯π‘˜,π‘₯π‘˜βˆ’1,…,π‘₯𝑗π‘₯∢=πœ†π‘˜,π‘₯π‘˜βˆ’1,…,π‘₯𝑗+1ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ†π‘˜,π‘₯π‘˜βˆ’1,…,π‘₯𝑗+1π‘₯𝑗(2.1) for any 𝑗=π‘˜βˆ’1,π‘˜βˆ’2,…,1.

Proposition 2.2 (see [5]). Let ℀𝑛 be the cyclic group of order 𝑛=𝑝𝛽11𝑝𝛽22β‹―π‘π›½π‘˜π‘˜,(2.2) where 𝑝1,…,π‘π‘˜ are distinct prime numbers and 𝛽1,…,π›½π‘˜ are positive integers. Then the number 𝑅(℀𝑛) of rooted chains of subgroups in the lattice of subgroups of ℀𝑛 is the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ.(2.3)

Let β„€ be the set of all integer numbers. Given distinct positive integers 𝑖1,…,𝑖𝑑, we define a function πœ‹π‘–1β‹―π‘–π‘‘βˆΆβ„€π‘˜βŸΌβ„€π‘˜,ξ€·π‘₯1,…,π‘₯π‘˜ξ€ΈβŸΌξ€·π‘¦1,…,π‘¦π‘˜ξ€Έ,(2.4) where 𝑦ℓ=ξ‚»π‘₯β„“,ifℓ≠𝑖𝑗π‘₯βˆ€π‘—=1,…,𝑑,β„“βˆ’1,β„“=𝑖𝑗forsome𝑗suchthat1≀𝑗≀𝑑.(2.5)

Most of our notations are standard and for undefined group theoretical terminologies we refer the reader to [6, 7]. For a general theory of solving a recurrence relation using a generating function, we refer the reader to [8, 9].

3. The Number of Chains of Subgroups of the Dicyclic Group 𝐡4𝑛

Throughout the section, we assume that π‘›βˆΆ=𝑝𝛽11𝑝𝛽22β‹―π‘π›½π‘˜π‘˜,(3.1) is a positive integer, where 𝑝1,…,π‘π‘˜ are distinct prime numbers and 𝛽1,…,π›½π‘˜ are nonnegative integers and the dicyclic group 𝐡4𝑛 of order 4𝑛 is defined by the following presentation: 𝐡4π‘›ξ«βˆΆ=π‘Ž,π‘βˆ£π‘Ž2𝑛=𝑒,𝑏2=π‘Žπ‘›,π‘π‘Žπ‘βˆ’1=π‘Žβˆ’1,(3.2) where 𝑒 is the identity element.

By the elementary group theory, the following is wellknown.

Lemma 3.1. The dicyclic group 𝐡4𝑛 has an index 2 subgroup βŸ¨π‘ŽβŸ©, which is isomorphic to β„€2𝑛, and has 𝑝𝑖 index 𝑝𝑖 subgroups βŸ¨π‘Žπ‘π‘–,π‘βŸ©,βŸ¨π‘Žπ‘π‘–ξ«π‘Ž,π‘Žπ‘βŸ©,…,𝑝𝑖,π‘Žπ‘π‘–βˆ’1𝑏,(3.3) which are isomorphic to the dicyclic group 𝐡4𝑛/𝑝𝑖 of order 4𝑛/𝑝𝑖 where 𝑖=1,2,…,π‘˜.

Lemma 3.2. (1) For any 𝑖=1,2,…,π‘˜, βŸ¨π‘Žπ‘π‘–,π‘Žπ‘Ÿπ‘βŸ©βˆ©βŸ¨π‘Žπ‘π‘–,π‘Žπ‘ π‘βŸ©=βŸ¨π‘Žπ‘π‘–βŸ©β‰…β„€2𝑛/𝑝𝑖,(3.4) where 0β‰€π‘Ÿ<π‘ β‰€π‘π‘–βˆ’1.
(2) For any distinct prime factors 𝑝𝑖1,𝑝𝑖2,…,𝑝𝑖𝑑 of 𝑛, βŸ¨π‘Žπ‘π‘–1,π‘Žπ‘Ÿ1π‘βŸ©βˆ©βŸ¨π‘Žπ‘π‘–2,π‘Žπ‘Ÿ2π‘βŸ©βˆ©β‹―βˆ©βŸ¨π‘Žπ‘π‘–π‘‘,π‘Žπ‘Ÿπ‘‘π‘βŸ©β‰…π΅4𝑛/𝑝𝑖1⋯𝑝𝑖𝑑,(3.5) where π‘Ÿ1,…,π‘Ÿπ‘‘ are nonnegative integers.

Proof. (1) To the contrary suppose that βŸ¨π‘Žπ‘π‘–,π‘Žπ‘Ÿπ‘βŸ©βˆ©βŸ¨π‘Žπ‘π‘–,π‘Žπ‘ π‘βŸ©β‰ βŸ¨π‘Žπ‘π‘–βŸ©.(3.6) Then π‘Žπ‘π‘–π‘’+π‘Ÿπ‘=π‘Žπ‘π‘–π‘£+𝑠𝑏 for some integers 𝑒 and 𝑣. This implies π‘π‘–βˆ£π‘ βˆ’π‘Ÿ. Since 0β‰€π‘Ÿ<π‘ β‰€π‘π‘–βˆ’1, we have 𝑠=π‘Ÿ, a contradiction.
(2) We only give its proof when 𝑑=2. The general case can be proved by the inductive process. Let 𝐾∢=βŸ¨π‘Žπ‘π‘–1,π‘Žπ‘Ÿ1π‘βŸ©βˆ©βŸ¨π‘Žπ‘π‘–2,π‘Žπ‘Ÿ2π‘βŸ©.(3.7) Clearly, π‘Žπ‘π‘–1𝑝𝑖2∈𝐾. Since gcd(𝑝𝑖1,𝑝𝑖2)=1, there exist integers 𝑒 and 𝑣 such that 𝑝𝑖1𝑒+𝑝𝑖2𝑣=1. Note that π‘Žπ‘π‘–1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))+π‘Ÿ1𝑏=π‘Žπ‘π‘–1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))π‘Žπ‘Ÿ1π‘βˆˆβŸ¨π‘Žπ‘π‘–1,π‘Žπ‘Ÿ1π‘βŸ©. On the other hand, π‘Žπ‘π‘–1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))+π‘Ÿ1𝑏=π‘Žβˆ’π‘π‘–1𝑒(π‘Ÿ1βˆ’π‘Ÿ2)+π‘Ÿ1𝑏=π‘Žπ‘π‘–2𝑣(π‘Ÿ1βˆ’π‘Ÿ2)βˆ’(π‘Ÿ1βˆ’π‘Ÿ2)+π‘Ÿ1𝑏since𝑝𝑖1𝑒+𝑝𝑖2𝑣=1=π‘Žπ‘π‘–2𝑣(π‘Ÿ1βˆ’π‘Ÿ2)+π‘Ÿ2π‘βˆˆβŸ¨π‘Žπ‘π‘–2,π‘Žπ‘Ÿ2π‘βŸ©.(3.8) Considering the order of 𝐾, one can see that 𝐾=βŸ¨π‘Žπ‘π‘–1𝑝𝑖2,π‘Žπ‘π‘–1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))+π‘Ÿ1π‘βŸ©. Since ξ€·π‘Žπ‘π‘–1𝑝𝑖2ξ€Έ4𝑛/𝑝𝑖1𝑝𝑖2ξ€·π‘Ž=𝑒,𝑝𝑖1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))+π‘Ÿ1𝑏2=𝑏2=π‘Žπ‘›=ξ€·π‘Žπ‘π‘–1𝑝𝑖2𝑛/𝑝𝑖1𝑝𝑖2,ξ€·π‘Žπ‘π‘–1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))+π‘Ÿ1π‘π‘Žξ€Έξ€·π‘π‘–1𝑝𝑖2π‘Žξ€Έξ€·π‘π‘–1(βˆ’π‘’(π‘Ÿ1βˆ’π‘Ÿ2))+π‘Ÿ1π‘ξ€Έβˆ’1=ξ€·π‘Žπ‘π‘–1𝑝𝑖2ξ€Έβˆ’1,(3.9) we have 𝐾≅𝐡4𝑛/𝑝𝑖1𝑝𝑖2.

By Lemma 3.1, we have 𝒰𝐡4𝑛=π’žβŸ¨π‘ŽβŸ©β‰…β„€2π‘›ξ€Έπ‘˜ξšπ‘π‘–=0π‘–βˆ’1ξšπ‘—=0π’žπ‘Žξ€·ξ«π‘π‘–,π‘Žπ‘—π‘ξ¬β‰…π΅4𝑛/𝑝𝑖.(3.10) Using the inclusion-exclusion principle and Lemma 3.2, one can see that the number π‘ˆ(𝐡4𝑛) has the following form: π‘ˆξ€·π΅4𝑛℀=𝐢2𝑛+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜π‘§π‘–1,…,𝑖𝑑𝐢℀2𝑛/𝑝𝑖1⋯𝑝𝑖𝑑+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜π‘π‘–1,…,𝑖𝑑𝐢𝐡4𝑛/𝑝𝑖1⋯𝑝𝑖𝑑(3.11) for suitable integers 𝑧𝑖1,…,𝑖𝑑 and 𝑏𝑖1,…,𝑖𝑑. In the following, we determine the numbers 𝑧𝑖1,…,𝑖𝑑 and 𝑏𝑖1,…,𝑖𝑑 explicitly.

Lemma 3.3. (1) 𝑏𝑖1,𝑖2,…,𝑖𝑑=(βˆ’1)𝑑+1𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑.
(2) 𝑧𝑖1,𝑖2,…,𝑖𝑑=(βˆ’1)𝑑𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑.

Proof. (1) Clearly 𝑏𝑖1=(βˆ’1)1+1𝑝𝑖1=𝑝𝑖1 for any 𝑖1=1,…,π‘˜. For any integer 𝑑β‰₯2, one can see by Lemma 3.2 that among intersections of the subgroups of the right-hand side of (3.10), the group isomorphic to 𝐡4𝑛/𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑 only appears in 𝑑-intersection of the subgroups ξ«π‘Žπ‘π‘–1,π‘Žπ‘—1𝑏,ξ«π‘Žπ‘π‘–2,π‘Žπ‘—2π‘ξ¬ξ«π‘Ž,…,𝑝𝑖𝑑,π‘Žπ‘—π‘‘π‘ξ¬,(3.12) where 0β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿβˆ’1 and 1β‰€π‘Ÿβ‰€π‘‘. Since there are 𝑝𝑖11𝑝𝑖21⋯𝑝𝑖𝑑1ξ€Έ=𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑 such choices, we have 𝑏𝑖1,𝑖2,…,𝑖𝑑=(βˆ’1)𝑑+1𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑.
(2) By Lemma 3.2, one can see that among intersections of the subgroups of the right-hand side of (3.10), the group isomorphic to β„€2𝑛/𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑 only appears one of the following two forms: ξ«π‘ŽβŸ¨π‘ŽβŸ©βˆ©π‘π‘–1,π‘Žπ‘—1π‘ξ¬βˆ©ξ«π‘Žπ‘π‘–2,π‘Žπ‘—2π‘ξ¬ξ«π‘Žβˆ©β‹―βˆ©π‘π‘–π‘‘,π‘Žπ‘—π‘‘π‘ξ¬,ξ«π‘Žπ‘π‘–1,π‘Žπ‘—1π‘ξ¬βˆ©ξ«π‘Žπ‘π‘–2,π‘Žπ‘—2π‘ξ¬ξ«π‘Žβˆ©β‹―βˆ©π‘π‘–π‘‘,π‘Žπ‘—π‘‘π‘ξ¬,(3.13) where 0β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿβˆ’1 and 1β‰€π‘Ÿβ‰€π‘‘, and each subgroup type in the first form must appear at least once, and it can appear more than once, while each subgroup type in the second form must appear at least once, and one of the subgroup types must appear more than once. Let 𝛾 be the number of the groups isomorphic to β„€2𝑛/𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑 obtained from the first form, and let 𝛿 be the number of the groups isomorphic to β„€2𝑛/𝑝𝑖1𝑝𝑖2⋯𝑝𝑖𝑑 obtained from the second form. Then clearly 𝑧𝑖1,𝑖2,β‹―,𝑖𝑑=𝛾+𝛿. Note that 𝛾=𝑝𝑖1+β‹―+π‘π‘–π‘‘βˆ’π‘‘ξ“π‘˜=0(βˆ’1)𝑑+2+π‘˜ξ“π‘—1+β‹―+𝑗𝑑=𝑑+π‘˜,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘˜ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =ξ“π‘˜β‰₯0𝑗1+β‹―+𝑗𝑑=𝑑+π‘˜,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿ,1β‰€π‘Ÿβ‰€π‘‘(βˆ’1)𝑑𝑑+π‘˜ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿ,1β‰€π‘Ÿβ‰€π‘‘(βˆ’1)𝑗1+β‹―+π‘—π‘‘π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =π‘‘ξ‘π‘Ÿ=11β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿ(βˆ’1)𝑗1+β‹―+π‘—π‘‘βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =(βˆ’1)𝑑.(3.14) On the other hand, 𝛿=𝑝𝑖1+β‹―+π‘π‘–π‘‘βˆ’π‘‘βˆ’1ξ“π‘˜=0(βˆ’1)𝑑+2+π‘˜ξ“π‘—1+β‹―+𝑗𝑑=𝑑+1+π‘˜,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =𝑝𝑖1+β‹―+π‘π‘–π‘‘βˆ’π‘‘βˆ’1ξ“π‘˜=0(βˆ’1)𝑑+2+π‘˜ξ“π‘—1+β‹―+𝑗𝑑=𝑑+1+π‘˜,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ +(βˆ’1)𝑑+1𝑗1+β‹―+𝑗𝑑=𝑑,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ βˆ’(βˆ’1)𝑑+1𝑗1+β‹―+𝑗𝑑=𝑑,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =𝑝𝑖1+β‹―+π‘π‘–π‘‘βˆ’π‘‘ξ“π‘˜=0(βˆ’1)𝑑+1+π‘˜ξ“π‘—1+β‹―+𝑗𝑑=𝑑+π‘˜,1β‰€π‘—π‘Ÿβ‰€π‘π‘—π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ βˆ’(βˆ’1)𝑑+1𝑗1+β‹―+𝑗𝑑=𝑑,1β‰€π‘—π‘Ÿβ‰€π‘π‘–π‘Ÿπ‘‘,1β‰€π‘Ÿβ‰€π‘‘ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘π‘–π‘Ÿπ‘—π‘ŸβŽžβŽŸβŽŸβŽ =(βˆ’1)π‘‘βˆ’(βˆ’1)𝑑+1𝑝𝑖1⋯𝑝𝑖𝑑.(3.15) Therefore, we have 𝑧𝑖1,𝑖2,…,𝑖𝑑=(βˆ’1)𝑑𝑝𝑖1⋯𝑝𝑖𝑑.

By Proposition 2.1 and Lemma 3.3, (3.11) becomes 𝑅𝐡4𝑛℀=2𝑅2𝑛+21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜(βˆ’1)𝑑𝑝𝑖1⋯𝑝𝑖𝑑𝑅℀2𝑛/𝑝𝑖1⋯𝑝𝑖𝑑+21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜(βˆ’1)𝑑+1𝑝𝑖1⋯𝑝𝑖𝑑𝑅𝐡4𝑛/𝑝𝑖1⋯𝑝𝑖𝑑.(3.16) Let π‘Žπ›½1,…,π›½π‘˜βˆΆ=𝑅(𝐡4𝑛) and let 𝑏𝛽1,…,π›½π‘˜βˆΆ=𝑅(β„€2𝑛). Then (3.16) becomes π‘Žπ›½1,…,π›½π‘˜=2𝑏𝛽1,…,π›½π‘˜ξ“+21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘π‘πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)+21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘π‘Žπœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜).(3.17)

Throughout the remaining part of the section, we solve the recurrence relation of (3.17) by using generating function technique. From now on, we allow each 𝛽𝑖 to be zero for computational convenience.

Let πœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,π‘₯π‘˜βˆ’1,…,π‘₯π‘—ξ€ΈβˆΆ=βˆžξ“π›½π‘—=0β‹―βˆžξ“π›½π‘˜βˆ’1∞=0ξ“π›½π‘˜=0π‘Žπ›½1,…,π›½π‘˜π‘₯π›½π‘˜π‘˜π‘₯π›½π‘˜βˆ’1π‘˜βˆ’1β‹―π‘₯𝛽𝑗𝑗,πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,π‘₯π‘˜βˆ’1,…,π‘₯π‘—ξ€ΈβˆΆ=βˆžξ“π›½π‘—=0β‹―βˆžξ“π›½π‘˜βˆ’1∞=0ξ“π›½π‘˜=0𝑏𝛽1,…,π›½π‘˜π‘₯π›½π‘˜π‘˜π‘₯π›½π‘˜βˆ’1π‘˜βˆ’1β‹―π‘₯𝛽𝑗𝑗,(3.18) where 𝑗=π‘˜,π‘˜βˆ’1,…,1.

For a fixed integer 𝑛=𝑝𝛽11𝑝𝛽22β‹―π‘π›½π‘˜π‘˜ such that 𝑝1,…,π‘π‘˜ are distinct prime numbers and 𝛽1,…,π›½π‘˜ are non-negative integers, we define a function πœ‡ as follows. πœ‡ξ€·π‘₯π‘˜ξ€ΈβˆΆ=1βˆ’2π‘π‘˜π‘₯π‘˜,πœ‡ξ€·π‘₯π‘˜,…,π‘₯𝑗π‘₯∢=πœ‡π‘˜,…,π‘₯𝑗+1ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗+1𝑝𝑗π‘₯𝑗(3.19) for any 𝑗=π‘˜βˆ’1,π‘˜βˆ’2,…,1.

Lemma 3.4. Let π‘˜ be a positive integer. If π‘˜=1, then πœ‡ξ€·π‘₯1ξ€Έπœ“π›½1ξ€·π‘₯1ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡1πœ™ξ€Έξ€Έπ›½1ξ€·π‘₯1ξ€Έ.(3.20) If π‘˜β‰₯2, then πœ‡ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯𝑗=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯π‘—Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£πœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯𝑗+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’1,1β‰€π‘‘β‰€π‘—βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’1,1β‰€π‘‘β‰€π‘—βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯⎦(3.21) for any 𝑗=π‘˜,π‘˜βˆ’1,…,2.

Proof. Assume first that π‘˜=1. Then (3.17) with π‘˜=1 gives us that π‘Žπ›½1=2𝑏𝛽1+2𝑝1π‘Žπ›½1βˆ’1βˆ’2𝑝1𝑏𝛽1βˆ’1.(3.22) Taking βˆ‘βˆžπ›½1=1 to both sides of (3.22), we have ξ€·1βˆ’2𝑝1π‘₯1ξ€Έπœ“π›½1ξ€·π‘₯1ξ€Έ=ξ€·2βˆ’2𝑝1π‘₯1ξ€Έπœ™π›½1ξ€·π‘₯1ξ€Έ(3.23) because π‘Ž0=𝑅(𝐡4𝑝01)=𝑅(β„€22)=22 and 𝑏0=𝑅(β„€2𝑝01)=𝑅(β„€2)=2 by a direct computation.
From now on, we assume that π‘˜β‰₯2. We prove (3.21) by double induction on π‘˜ and 𝑗. Equation (3.17) with π‘˜=2 gives us that π‘Žπ›½1,𝛽2=2𝑏𝛽1,𝛽2βˆ’2𝑝1𝑏𝛽1βˆ’1,𝛽2βˆ’2𝑝2𝑏𝛽1,𝛽2βˆ’1+2𝑝1𝑝2𝑏𝛽1βˆ’1,𝛽2βˆ’1+2𝑝1π‘Žπ›½1βˆ’1,𝛽2+2𝑝2π‘Žπ›½1,𝛽2βˆ’1βˆ’2𝑝1𝑝2π‘Žπ›½1βˆ’1,𝛽2βˆ’1.(3.24) Taking βˆ‘βˆžπ›½2=1π‘₯𝛽22 of both sides of (3.24), we have ξ€·1βˆ’2𝑝2π‘₯2ξ€Έπœ“π›½1,𝛽2ξ€·π‘₯2ξ€Έ=ξ€·2βˆ’2𝑝2π‘₯2𝑝1πœ“π›½1βˆ’1,𝛽2ξ€·π‘₯2ξ€Έ+πœ™π›½1,𝛽2ξ€·π‘₯2ξ€Έβˆ’π‘1πœ™π›½1βˆ’1,𝛽2ξ€·π‘₯2ξ€Έξ€»(3.25) because π‘Žπ›½1,0=π‘Žπ›½1 and 𝑏𝛽1,0=𝑏𝛽1 by the definition, and π‘Žπ›½1,0βˆ’2𝑏𝛽1,0βˆ’2𝑝1π‘Žπ›½1,0+2𝑝1𝑏𝛽1βˆ’1,0=0(3.26) by (3.17) with π‘˜=1. That is, πœ‡ξ€·π‘₯2ξ€Έπœ“π›½1,𝛽2ξ€·π‘₯2ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡2πœ™ξ€Έξ€Έξ€Ίπ›½1,𝛽2ξ€·π‘₯2ξ€Έβˆ’π‘1πœ™π›½1βˆ’1,𝛽2ξ€·π‘₯2ξ€Έ+𝑝1πœ“π›½1βˆ’1,𝛽2ξ€·π‘₯2.ξ€Έξ€»(3.27) Thus (3.21) holds for π‘˜=2.
Assume now that (3.21) holds from 2 to π‘˜βˆ’1 and consider the case for π‘˜. Note that the last two terms of the right-hand side of (3.17) can be divided into three terms, respectively, as follows: 21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘π‘πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)=βˆ’2π‘π‘˜π‘π›½1,…,π›½π‘˜βˆ’1,π›½π‘˜βˆ’1βˆ’2π‘π‘˜ξ“1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘π‘πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,π›½π‘˜βˆ’1)+21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘π‘πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜),21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜,1β‰€π‘‘β‰€π‘˜(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘π‘Žπœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)=2π‘π‘˜π‘Žπ›½1,…,π›½π‘˜βˆ’1,π›½π‘˜βˆ’1βˆ’2π‘π‘˜ξ“1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘π‘Žπœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,π›½π‘˜βˆ’1)+21≀𝑖1<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘π‘Žπœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜).(3.28) Taking βˆ‘βˆžπ›½π‘˜=1π‘₯π›½π‘˜π‘˜ of both sides of (3.17) and using (3.28), one can see that ξ€·1βˆ’2π‘π‘˜π‘₯π‘˜ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜ξ€Έ=ξ€·2βˆ’2π‘π‘˜π‘₯π‘˜ξ€ΈΓ—βŽ‘βŽ’βŽ’βŽ’βŽ£πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜ξ€Έ+1≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,π›½π‘˜)ξ€·π‘₯π‘˜ξ€Έ+1≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯⎦+π‘Žπ›½1,…,π›½π‘˜βˆ’1,0βˆ’2𝑏𝛽1,…,π›½π‘˜βˆ’1,0ξ“βˆ’21≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘π‘πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,0)ξ“βˆ’21≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘π‘Žπœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,0).(3.29) Further since π‘Žπ›½1,…,π›½π‘˜βˆ’1,0βˆ’2𝑏𝛽1,…,π›½π‘˜βˆ’1,0ξ“βˆ’21≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘π‘πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,0)ξ“βˆ’21≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘π‘Žπœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,0)=0(3.30) by (3.17), we have ξ€·1βˆ’2π‘π‘˜π‘₯π‘˜ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜ξ€Έ=ξ€·2βˆ’2π‘π‘˜π‘₯π‘˜ξ€ΈΓ—βŽ‘βŽ’βŽ’βŽ’βŽ£πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜ξ€Έ+1≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜βˆ’1,π›½π‘˜)ξ€·π‘₯π‘˜ξ€Έ+1≀𝑖<β‹―<π‘–π‘‘β‰€π‘˜βˆ’1,1β‰€π‘‘β‰€π‘˜βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯⎦.(3.31) Thus (3.21) holds for 𝑗=π‘˜. Assume that (3.21) holds from π‘˜ to 𝑗 and consider the case for π‘—βˆ’1. Note that the last two terms of the right-hand side of (3.21) can be divided into three terms, respectively, as follows: 1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’1,1β‰€π‘‘β‰€π‘—βˆ’1(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗=βˆ’π‘π‘—βˆ’1πœ™π›½1,…,π›½π‘—βˆ’2,π›½π‘—βˆ’1βˆ’1,𝛽𝑗,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έβˆ’π‘π‘—βˆ’11≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘—βˆ’2,π›½π‘—βˆ’1βˆ’1,𝛽𝑗,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗,(3.32)1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’1,1β‰€π‘‘β‰€π‘—βˆ’1(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗=π‘π‘—βˆ’1πœ“π›½1,…,π›½π‘—βˆ’2,π›½π‘—βˆ’1βˆ’1,𝛽𝑗,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έβˆ’π‘π‘—βˆ’11≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘—βˆ’2,π›½π‘—βˆ’1βˆ’1,𝛽𝑗,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗.(3.33) Taking βˆ‘βˆžπ›½π‘—βˆ’1=1π‘₯π›½π‘—βˆ’1π‘—βˆ’1 of both sides of (3.21), we have πœ‡ξ€·π‘₯π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘—βˆ’1,…,π‘₯π‘˜ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£πœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘—βˆ’1,…,π‘₯π‘˜ξ€Έ+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1ξ€Έ+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ¦ξ€·π‘₯+πœ‡π‘˜,…,π‘₯π‘—ξ€Έπœ“π›½1,…,π›½π‘—βˆ’2,0,𝛽𝑗,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘—βˆ’2,0,𝛽𝑗,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘—βˆ’2,0,𝛽𝑗,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗.(3.34) Note that πœ‡ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έπœ“π›½1,…,π›½π‘—βˆ’2,0,𝛽𝑗,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘—βˆ’2,0,𝛽𝑗,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯π‘—ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘—βˆ’2,0,𝛽𝑗,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗=0(3.35) by induction hypothesis. Thus πœ‡ξ€·π‘₯π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘—βˆ’1,…,π‘₯π‘˜ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£πœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘—βˆ’1,…,π‘₯π‘˜ξ€Έ+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑𝑝𝑖1β‹―π‘π‘–π‘‘πœ™πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1ξ€Έ+1≀𝑖1<β‹―<π‘–π‘‘β‰€π‘—βˆ’2,1β‰€π‘‘β‰€π‘—βˆ’2(βˆ’1)𝑑+1𝑝𝑖1β‹―π‘π‘–π‘‘πœ“πœ‹π‘–1𝑑⋯𝑖(𝛽1,…,π›½π‘˜)ξ€·π‘₯π‘˜,…,π‘₯𝑗,π‘₯π‘—βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯⎦.(3.36) Therefore, (3.21) holds for π‘—βˆ’1.

Equation (3.21) with 𝑗=2 gives us that πœ‡ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯2Γ—ξ€Ίπœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έβˆ’π‘1πœ™π›½1βˆ’1,𝛽2,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έ+𝑝1πœ“π›½1βˆ’1,𝛽2,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2.ξ€Έξ€»(3.37) Taking βˆ‘βˆžπ›½1=1π‘₯𝛽11 of both sides of (3.37), we get that πœ‡ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2,π‘₯1ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯1πœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2,π‘₯1ξ€Έξ€·π‘₯+πœ‡π‘˜,…,π‘₯2ξ€Έπœ“0,𝛽2,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯2πœ™ξ€Έξ€Έ0,𝛽2,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έ.(3.38)

Lemma 3.5. If π‘˜β‰₯2, then πœ‡ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έπœ“0,𝛽2,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯2πœ™ξ€Έξ€Έ0,𝛽2,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έ.(3.39)

Proof. If π‘˜=2, then since πœ“0,𝛽2(π‘₯2)=πœ“π›½2(π‘₯2) and πœ™0,𝛽2(π‘₯2)=πœ™π›½2(π‘₯2), the equation πœ‡ξ€·π‘₯2ξ€Έπœ“0,𝛽2ξ€·π‘₯2ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡2πœ™ξ€Έξ€Έ0,𝛽2ξ€·π‘₯2ξ€Έ(3.40) holds by (3.20). Assume now that (3.39) holds for π‘˜. Then by (3.38) we get that πœ‡ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2,π‘₯1ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯1πœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯2,π‘₯1ξ€Έ,(3.41) which implies that πœ‡ξ€·π‘₯π‘˜+1,…,π‘₯2ξ€Έπœ“0,𝛽2,…,π›½π‘˜+1ξ€·π‘₯π‘˜+1,…,π‘₯2ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜+1,…,π‘₯2πœ™ξ€Έξ€Έ0,𝛽2,…,π›½π‘˜+1ξ€·π‘₯π‘˜+1,…,π‘₯2ξ€Έ.(3.42) Thus (3.39) holds for π‘˜+1.

By Lemmas 3.4 and 3.5 and (3.38), we have πœ‡ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έπœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=ξ€·ξ€·π‘₯1+πœ‡π‘˜,…,π‘₯1πœ™ξ€Έξ€Έπ›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ.(3.43) We now need to find the function πœ™π›½1,…,π›½π‘˜(π‘₯π‘˜,…,π‘₯1) explicitly.

Lemma 3.6. If 𝑝1=2, then πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩2πœ†ξ€·π‘₯11,π‘–π‘“π‘˜=1,1+πœ†ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έξƒ­1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ,π‘–π‘“π‘˜β‰₯2.(3.44) If 𝑝𝑖≠2 for 𝑖=1,2,…,π‘˜, then πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=11+πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έξƒ­1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ.(3.45)

Proof. We first assume that 𝑝1=2. Then by Proposition 2.2, 𝑏𝛽1,𝛽2,…,π›½π‘˜ξ‚€β„€=𝑅𝑝𝛽11+1𝑝𝛽22𝑝𝛽33β‹―π‘π›½π‘˜π‘˜ξ‚(3.46) is the coefficient of π‘₯𝛽11+1π‘₯𝛽22π‘₯𝛽33β‹―π‘₯π›½π‘˜π‘˜ of 1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ,(3.47) which implies that 𝑏𝛽1,𝛽2,…,π›½π‘˜ is the coefficient of π‘₯𝛽11π‘₯𝛽22π‘₯𝛽33β‹―π‘₯π›½π‘˜π‘˜ of 2πœ†ξ€·π‘₯11ifπ‘˜=1,1+πœ†ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έξƒ­1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έifπ‘˜β‰₯2,(3.48) and hence by the definition of πœ™ we get that πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩2πœ†ξ€·π‘₯11,ifπ‘˜=1,1+πœ†ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έξƒ­1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ,ifπ‘˜β‰₯2.(3.49)
Assume now that 𝑝𝑖≠2 for 𝑖=1,2,…,π‘˜. Since 𝑏𝛽1,…,π›½π‘˜=𝑅(β„€2𝑝𝛽11𝑝𝛽22β‹―π‘π›½π‘˜π‘˜), by Proposition 2.2𝑏𝛽1,…,π›½π‘˜ is the coefficient of π‘₯11π‘₯𝛽12π‘₯𝛽23β‹―π‘₯π›½π‘˜π‘˜+1 of 1πœ†ξ€·π‘₯π‘˜+1,…,π‘₯1ξ€Έ.(3.50) Since 1πœ†ξ€·π‘₯π‘˜+1,…,π‘₯1ξ€Έ=1πœ†ξ€·π‘₯π‘˜+1,…,π‘₯2ξ€Έβˆ’ξ€·ξ€·π‘₯1+πœ†π‘˜+1,…,π‘₯2π‘₯ξ€Έξ€Έ1=1πœ†ξ€·π‘₯π‘˜+1,…,π‘₯2ξ€Έ1ξ€Ίξ€·ξ€·π‘₯1βˆ’1+1/πœ†π‘˜+1,…,π‘₯2π‘₯ξ€Έξ€Έξ€»1(3.51) by the definition, 𝑏𝛽1,…,π›½π‘˜ is the coefficient of π‘₯𝛽12π‘₯𝛽23β‹―π‘₯π›½π‘˜π‘˜+1 of 1πœ†ξ€·π‘₯π‘˜+1,…,π‘₯211+πœ†ξ€·π‘₯π‘˜+1,…,π‘₯2ξ€Έξƒ­.(3.52) By changing the variables π‘₯2,π‘₯3,…,π‘₯π‘˜+1 by π‘₯1,π‘₯2,…,π‘₯π‘˜, respectively, we get that 𝑏𝛽1,…,π›½π‘˜ is the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of 1πœ†ξ€·π‘₯π‘˜,…,π‘₯111+πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έξƒ­.(3.53) By the definition of πœ™π›½1,…,π›½π‘˜(π‘₯π‘˜,…,π‘₯1), we have πœ™π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=1πœ†ξ€·π‘₯π‘˜,…,π‘₯111+πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έξƒ­.(3.54)

By Proposition 2.1, (3.43), and Lemma 3.6, we have the following theorem.

Theorem 3.7. Let π‘›βˆΆ=𝑝𝛽11𝑝𝛽22β‹―π‘π›½π‘˜π‘˜,(3.55) be a positive integer such that 𝑝1,…,π‘π‘˜ are distinct prime numbers and 𝛽1,…,π›½π‘˜ are positive integers. Let 𝐡4π‘›ξ«βˆΆ=π‘Ž,π‘βˆ£π‘Ž2𝑛=𝑒,𝑏2=π‘Žπ‘›,π‘π‘Žπ‘βˆ’1=π‘Žβˆ’1(3.56) be the dicyclic group of order 4𝑛. Let 𝑅(𝐡4𝑛) be the number of rooted chains of subgroups in the lattice of subgroups of 𝐡4𝑛. (1)If 𝑝1=2, then 𝑅(𝐡4𝑛) is the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of πœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξƒ¬11+πœ‡ξ€·π‘₯1ξ€Έξƒ­2πœ†ξ€·π‘₯11,π‘–π‘“π‘˜=1,1+πœ‡ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ11+πœ†ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έξƒ­1πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ,π‘–π‘“π‘˜β‰₯2.(3.57)(2)If 𝑝𝑖≠2 for 𝑖=1,2,…,π‘˜, then 𝑅(𝐡4𝑛) is the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of πœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=11+πœ‡ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έξƒ­1πœ†ξ€·π‘₯π‘˜,…,π‘₯111+πœ†ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έξƒ­.(3.58)
Furthermore, the number 𝐢(𝐡4𝑛) of chains of subgroups in the lattice of subgroups of 𝐡4𝑛 is the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of 2πœ“π›½1,…,π›½π‘˜ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έβˆ’π‘˜ξ‘π‘–=111βˆ’π‘₯𝑖.(3.59)

We now want to find the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of πœ“π›½1,…,π›½π‘˜(π‘₯π‘˜,…,π‘₯1) explicitly. Since 1πœ‡ξ€·π‘₯π‘˜,…,π‘₯1ξ€Έ=1πœ‡ξ€·π‘₯π‘˜,…,π‘₯2ξ€Έ1ξ€Ίξ€·ξ€·π‘₯1βˆ’1+1/πœ‡π‘˜,…,π‘₯2𝑝1π‘₯1,(3.60) by the definition, the coefficient of π‘₯𝛽11 of 1/πœ‡(π‘₯π‘˜,…,π‘₯1) is 1πœ‡ξ€·π‘₯π‘˜,…,π‘₯211+πœ‡ξ€·π‘₯π‘˜,…,π‘₯2𝛽1𝑝𝛽11=𝑝𝛽11𝛽1𝑖1=0βŽ›βŽœβŽœβŽπ›½1𝑖1βŽžβŽŸβŽŸβŽ ξƒ¬1πœ‡ξ€·π‘₯π‘˜,…,π‘₯2𝑖1+1=𝑝𝛽11𝛽1𝑖1=0βŽ›βŽœβŽœβŽπ›½1𝑖1βŽžβŽŸβŽŸβŽ ξƒ¬1πœ‡ξ€·π‘₯π‘˜,…,π‘₯3𝑖1+11ξ€Ίξ€·ξ€·π‘₯1βˆ’1+1/πœ‡π‘˜,…,π‘₯3𝑝2π‘₯2𝑖1+1.(3.61) Thus the coefficient of π‘₯𝛽11π‘₯𝛽22 of 1/πœ‡(π‘₯π‘˜,…,π‘₯1) is 𝑝𝛽11𝑝𝛽22𝛽1𝑖1=0βŽ›βŽœβŽœβŽπ›½1𝑖1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘–1+𝛽2𝛽2βŽžβŽŸβŽŸβŽ ξƒ¬1πœ‡ξ€·π‘₯π‘˜,…,π‘₯3𝑖1+111+πœ‡ξ€·π‘₯π‘˜,…,π‘₯3𝛽2=𝑝𝛽11𝑝𝛽22𝛽1𝑖1𝛽=02𝑖2=0βŽ›βŽœβŽœβŽπ›½1𝑖1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½2𝑖2βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘–1+𝛽2𝛽2βŽžβŽŸβŽŸβŽ ξƒ¬1πœ‡ξ€·π‘₯π‘˜,…,π‘₯3𝑖1+𝑖2+1.(3.62) Continuing this process, one can see that the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of 1/πœ‡(π‘₯π‘˜,…,π‘₯1) is 2π›½π‘˜π‘π›½11𝑝𝛽22β‹―π‘π›½π‘˜π‘˜π›½1𝑖1𝛽=02𝑖2=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1⎞⎟⎟⎟⎠.(3.63) Similarly one can see that the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of 1/πœ†(π‘₯π‘˜,…,π‘₯1) is 2π›½π‘˜π›½1𝑖1𝛽=02𝑖2=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1⎞⎟⎟⎟⎠,(3.64) the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of [1+(1/πœ†(π‘₯π‘˜,…,π‘₯2))](1/πœ†(π‘₯π‘˜,…,π‘₯1)) is 2π›½π‘˜π›½1+1𝑖1𝛽=02𝑖2𝛽=03𝑖3=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0βŽ›βŽœβŽœβŽπ›½1𝑖+11βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½2+𝑖1𝛽2βŽžβŽŸβŽŸβŽ π‘˜βˆ’1ξ‘π‘Ÿ=2βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1⎞⎟⎟⎟⎠(3.65) and the coefficient of π‘₯𝛽11π‘₯𝛽22β‹―π‘₯π›½π‘˜π‘˜ of 1/πœ†(π‘₯π‘˜,…,π‘₯1)2 is 𝛽1ξ€Έ2+1π›½π‘˜π›½1𝑖1𝛽=02𝑖2=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½+1π‘Ÿ+1⎞⎟⎟⎟⎠.(3.66)

Therefore, one can have the following.

Corollary 3.8. Let 𝑛 and 𝐡4𝑛 be the positive integer and the dicyclic group, respectively, defined in Theorem 3.7. Let 𝑅(𝐡4𝑛) be the number of rooted chains of subgroups in the lattice of subgroups of 𝐡4𝑛. (1)If 𝑝1=2, then 𝑅𝐡4𝑛=2π›½π‘˜π›½1+1𝑖1𝛽=02𝑖2𝛽=03𝑖3=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0βŽ›βŽœβŽœβŽπ›½1𝑖+11βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½2+𝑖1𝛽2βŽžβŽŸβŽŸβŽ π‘˜βˆ’1ξ‘π‘Ÿ=2βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1⎞⎟⎟⎟⎠+2π›½π‘˜π›½1𝑗1𝛽=02𝑗2=0β‹―π›½π‘˜ξ“π‘—π‘˜=0βŽ‘βŽ’βŽ’βŽ’βŽ£βŽ‘βŽ’βŽ’βŽ’βŽ£π‘π‘—11𝑝𝑗22β‹―π‘π‘—π‘˜π‘˜π‘—1𝑖1𝑗=02𝑖2=0β‹―π‘—π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘—π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ‘—π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ‘—π‘Ÿ+1⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ’βŽ£π›½1βˆ’π‘—1+1𝑖1𝛽=02βˆ’π‘—2𝑖2𝛽=03βˆ’π‘—3𝑖3=0β‹―π›½π‘˜βˆ’1βˆ’π‘—π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0βŽ›βŽœβŽœβŽπ›½1βˆ’π‘—1𝑖+11βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½2βˆ’π‘—2+𝑖1𝛽2βˆ’π‘—2βŽžβŽŸβŽŸβŽ Γ—π‘˜βˆ’1ξ‘π‘Ÿ=2βŽ›βŽœβŽœβŽπ›½π‘Ÿβˆ’π‘—π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1βˆ’π‘—π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1βˆ’π‘—π‘Ÿ+1⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦⎀βŽ₯βŽ₯βŽ₯⎦,(3.67)where if π‘˜=1, then 𝑅(𝐡4β‹…2𝛽1)=22𝛽1+2 and if π‘˜=2, then 𝑅𝐡4β‹…2𝛽1𝑝𝛽22=2𝛽2𝛽1+1𝑖1=0βŽ›βŽœβŽœβŽπ›½1𝑖+11βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½2+𝑖1𝛽2⎞⎟⎟⎠+2𝛽2𝛽1𝑗1𝛽=02𝑗2=0⎑⎒⎒⎣⎑⎒⎒⎣2𝑗1𝑝𝑗22𝑗1𝑖1=0βŽ›βŽœβŽœβŽπ‘—1𝑖1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—2+𝑖1𝑗2⎞⎟⎟⎠⎀βŽ₯βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ’βŽ£π›½1βˆ’π‘—1+1𝑖1=0βŽ›βŽœβŽœβŽπ›½1βˆ’π‘—1𝑖+11βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½2βˆ’π‘—2+𝑖1𝛽2βˆ’π‘—2⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦⎀βŽ₯βŽ₯⎦.(3.68)(2)If 𝑝𝑖≠2 for 𝑖=1,2,…,π‘˜, then 𝑅𝐡4𝑛=2π›½π‘˜π›½1𝑖1𝛽=02𝑖2=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1⎞⎟⎟⎟⎠+𝛽1ξ€Έ2+1π›½π‘˜π›½1𝑖1𝛽=02𝑖2=0β‹―π›½π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1⎞⎟⎟⎟⎠+2π›½π‘˜π›½1𝑗1𝛽=02𝑗2=0β‹―π›½π‘˜ξ“π‘—π‘˜=0βŽ‘βŽ’βŽ’βŽ’βŽ£βŽ‘βŽ’βŽ’βŽ’βŽ£π‘π‘—11𝑝𝑗22β‹―π‘π‘—π‘˜π‘˜π‘—1𝑖1𝑗=02𝑖2=0β‹―π‘—π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘—π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ‘—π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ‘—π‘Ÿ+1⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£π›½1βˆ’π‘—1𝑖1𝛽=02βˆ’π‘—2𝑖2=0β‹―π›½π‘˜βˆ’1βˆ’π‘—π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿβˆ’π‘—π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1βˆ’π‘—π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1βˆ’π‘—π‘Ÿ+1⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦⎀βŽ₯βŽ₯βŽ₯⎦+2π›½π‘˜π›½1𝑗1𝛽=02𝑗2=0β‹―π›½π‘˜ξ“π‘—π‘˜=0βŽ‘βŽ’βŽ’βŽ’βŽ£βŽ‘βŽ’βŽ’βŽ’βŽ£π‘π‘—11𝑝𝑗22β‹―π‘π‘—π‘˜π‘˜π‘—1𝑖1𝑗=02𝑖2=0β‹―π‘—π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘—π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽπ‘—π‘Ÿ+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ‘—π‘Ÿ+1⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ’βŽ£ξ€·π›½1βˆ’π‘—1ξ€Έ+1𝛽1βˆ’π‘—1𝑖1𝛽=02βˆ’π‘—2𝑖2=0β‹―π›½π‘˜βˆ’1βˆ’π‘—π‘˜βˆ’1ξ“π‘–π‘˜βˆ’1=0π‘˜βˆ’1ξ‘π‘Ÿ=1βŽ›βŽœβŽœβŽπ›½π‘Ÿβˆ’π‘—π‘Ÿπ‘–π‘ŸβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽœβŽπ›½π‘Ÿ+1βˆ’π‘—π‘Ÿ+1+1+π‘Ÿβˆ‘π‘š=1π‘–π‘šπ›½π‘Ÿ+1βˆ’π‘—π‘Ÿ+1⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦⎀βŽ₯βŽ₯βŽ₯⎦,(3.69)where if π‘˜=1, then 𝑅𝐡4𝑝𝛽11=2𝛽1+𝛽1ξ€Έ2+1𝛽1+2𝛽1𝛽1𝑗1=0𝑝𝑗11+2𝛽1𝛽1𝑗1=0𝑝𝑗11𝛽1βˆ’π‘—1ξ€Έ+1=2𝛽1𝛽1𝑝+2+𝛽11+1βˆ’1𝑝1+π‘βˆ’1𝛽11+2βˆ’ξ€·π›½1𝑝+21+𝛽1+1𝑝1ξ€Έβˆ’12ξƒ­.(3.70)

Acknowledgments

The first author was funded by the Korean Government (KRF-2009-353-C00040). In the case of the third author, this research was supported by Basic Science Research Program Through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0025252).

References

  1. V. Murali and B. B. Makamba, β€œOn an equivalence of fuzzy subgroups. II,” Fuzzy Sets and Systems, vol. 136, no. 1, pp. 93–104, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. V. Murali and B. B. Makamba, β€œCounting the number of fuzzy subgroups of an abelian group of order pnqm,” Fuzzy Sets and Systems, vol. 144, no. 3, pp. 459–470, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. J.-M. Oh, β€œThe number of chains of subgroups of a finite cycle group,” European Journal of Combinatorics, vol. 33, no. 2, pp. 259–266, 2012. View at Publisher Β· View at Google Scholar
  4. M. Tărnăuceanu and L. Bentea, β€œOn the number of fuzzy subgroups of finite abelian groups,” Fuzzy Sets and Systems, vol. 159, no. 9, pp. 1084–1096, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. J. M. Oh, β€œThe number of chains of subgroups of the dihedral group,” Submitted.
  6. J. S. Rose, A Course on Group Theory, Dover Publications, New York, NY, USA, 1994.
  7. W. R. Scott, Group Theory, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
  8. M. Aigner, Combinatorial Theory, Springer, New York, NY, USA, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. A. Tucker, Applied Combinatorics, John Wiley & Sons, New York, NY, USA, 1995.