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Let $$\overline{ A}_\ell (n)$$ be the number of $$\ell $$ -regular overpartitions of n, i.e., overpartitions of n into parts not divisible by $$\ell $$ . Let $$\overline{ B}_\ell (n)$$ be the number of almost $$\ell $$ -regular overpartitions of n, i.e., overpartitions of n in which none of its overlined parts is divisible by $$\ell $$ . In this paper, we study the connections between $$\overline{ A}_3(n)$$ , respectively $$\overline{ B}_3(n)$$ , and the singular overpartition functions $$\overline{ C}_{12,5}(n)$$ and $$\overline{ C}_{12,1}(n)$$ which count the number of overpartitions into parts not divisible by 12 and in which only parts congruent to $$\pm 5 \pmod {12}$$ , respectively $$\pm 1 \pmod {12}$$ , may be overlined. We give a combinatorial proof for the the surprising identity $$\overline{ C}_{12,5}(n)=\overline{ C}_{12,1}(n-1)$$ . We also provide a linear homogeneous recurrence relation for $$\overline{ B}_3(n)$$ and give an alternate combinatorial interpretation for $$\overline{ B}_3(n)$$ .

Let $${\mathcal {O}}_r(n)$$ be the set of r-regular partitions of n, $${\mathcal {D}}_r(n)$$ the set of partitions of n with parts repeated at most $$r-1$$ times, $${\mathcal {O}}_{1,r}(n)$$ the set of partitions with exactly one part (possibly repeated) divisible by r, and let $${\mathcal {D}}_{1,r}(n)$$ be the set of partitions in which exactly one part appears at least r times. If $$E_{r, t}(n)$$ is the excess in the number of parts congruent to $$t \pmod r$$ in all partitions in $${\mathcal {O}}_r(n)$$ over the number of different parts appearing at least t times in all partitions in $${\mathcal {D}}_r(n)$$ , then $$E_{r, t}(n) = |{\mathcal {O}}_{1,r}(n)| = |{\mathcal {D}}_{1,r}(n)|$$ . We prove this analytically and combinatorially using a bijection due to Xiong and Keith. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in $$\mathcal {O}_r(n)$$ over the number of parts in all partitions in $$\mathcal {D}_r(n)$$ equals $$(r - 1)|\mathcal {O}_{1,r}(n)|$$ and also $$(r - 1)|\mathcal {D}_{1,r}(n)|$$ . Our work provides a new combinatorial proof of this result that does not use Glaisher’s bijection. We also give a new combinatorial proof based of the Xiong–Keith bijection for a second Beck-type identity that has been proved previously using Glaisher’s bijection.

The main result of this paper is an identity expressing the r-Stirling number of the first kind as a sum involving binomial coefficients and the Mobius function of the set-partition lattice. We provide three different proofs of this result: analytic, inductive, and combinatorial.

An alignment of a permutation $$\pi $$ on n letters is an ordered sequence of the disjoint cycles of $$\pi $$ . We consider several counting numbers related to alignments: the number of alignments of n and its mean, the number of cycles in all alignments of n and its mean, as well as the mean number of supernecklaces of type III (cycles of cycles) of n. We present a collection of identities relating these numbers and provide analytic and, with the exception of one identity, also combinatorial proofs of our results.

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities. In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.

Recently, Andrews and Merca considered the number of even parts in all partitions of n into distinct parts and obtained new combinatorial interpretations for this number. Their proofs rely on generating functions. In this paper, we provide purely combinatorial proofs of these results.

We prove several identities of the type ?$\alpha (n) = \Sigma_{k=0}^\infty \beta (\frac{(n - k(k + 1)/2)} {2})$?. Here, the functions ?$\alpha (n)$? and ?$\beta (n)$? count partitions with certain restrictions or the number of parts in certain partitions. Since G. N. Watson Proc. Lond. Math. Soc. (2) 42, 550-556 (1937) proved the identity for ?$\alpha (n) = Q(n)$?, the number of partitions of ?$n$? into distinct parts, and ?$\beta (n) = p(n)$?, Euler's partition function, we refer to these identities as Watson type identities. Our work is motivated by results of G. E. Andrews and M. Merca ''On the number of even parts in all partitions of $n$ into distinct parts'', Ann. Comb. (to appear) who recently discovered and proved new Euler type identities. We provide analytic proofs and explain how one could construct bijective proofs of our results. Dokažemo več identitet tipa ?$\alpha (n) = \Sigma_{k=0}^\infty \beta (\frac{(n - k(k + 1)/2)} {2})$?. Tukaj funkciji ?$\alpha (n)$? in ?$\beta (n)$? štejeta razčlenitve z določenimi omejitvami ali število delov v določenih razčlenitvah. Ker je Watson dokazal identiteto za ?$\alpha (n) = Q(n)$?, kjer je ?$Q(n)$? število razčlenitev števila ?$n$? na same različne dele, in za ?$\beta (n) = p(n)$?, kjer je ?$p(n)$? Eulerjeva razčlenitvena funkcija, tovrstne identitete imenujemo identitete Watsonovega tipa. Najino delo je motivirano z rezultati G. E. Andrewsa in drugega avtorja, ki je nedavno odkril in dokazal nove identitete Eulerjevega tipa. Podava analitične dokaze in razloživa, kako konstruirati bijektivne dokaze najinih rezultatov.

Let a(n) be the number of partitions of n, such that the set of even parts has exactly one element, b(n) be the difference between the number of parts in all odd partitions of n and the number of parts in all distinct partitions of n, and c(n) be the number of partitions of n in which exactly one part is repeated. Beck conjectured that a(n) = b(n) and Andrews, using generating functions, proved that a(n) = b(n) = c(n). We give a combinatorial proof of Andrews’ result. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. We prove combinatorially Beck’s second conjecture, which was also proved by Andrews using generating functions. Let c1(n) be the number of partitions of n, such that there is exactly one part occurring three times, while all other parts occur only once and let b1(n) be the difference between the total number of parts in the partitions of n into distinct parts and the total number of different parts in the partitions of n into odd parts. Then, c1(n) = b1(n).

An almost partition identity is an identity for partition numbers that is true asymptotically [Formula: see text] of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of [Formula: see text] is almost always equal to the number of partitions of [Formula: see text] in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of [Formula: see text] and the number of parts in all partitions of [Formula: see text] into distinct odd parts equals the number of partitions of [Formula: see text] in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

Partition identities are often statements asserting that the set \(\mathcal P_X\) of partitions of n subject to condition X is equinumerous to the set \(\mathcal P_Y\) of partitions of n subject to condition Y. A Beck-type identity is a companion identity to \(|\mathcal P_X|=|\mathcal P_Y|\) asserting that the difference b(n) between the number of parts in all partitions in \(\mathcal P_X\) and the number of parts in all partitions in \(\mathcal P_Y\) equals \(c|\mathcal P_{X'}|\) and also \(c|\mathcal P_{Y'}|\), where c is some constant related to the original identity, and \(X'\), respectively \(Y'\), is a condition on partitions that is a very slight relaxation of condition X, respectively Y. A second Beck-type identity involves the difference \(b'(n)\) between the total number of different parts in all partitions in \(\mathcal P_Y\) and the total number of different parts in all partitions in \(\mathcal P_X\). We extend these results to Beck-type identities accompanying all identities given by Euler pairs of order r (for any \(r\ge 2\)). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.

The minimal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is the smallest positive integer that is not a part of $\lambda$. For a positive integer $n$, $ \sigma\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $\sigma\, \rm{mex}(n)$. They showed, using generating functions, that $\sigma\, \rm{mex}(n)$ equals the number of partitions of $n$ into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function $\sigma\, \rm{mex}(n)$. We generalize this combinatorial interpretation to $\sigma_r\, \rm{mex}(n)$, the sum of least $r$-gaps in all partitions of $n$. The least $r$-gap of a partition $\lambda$ is the smallest positive integer that does not appear at least $r$ times as a part of $\lambda$., Comment: 15 pages; this version includes a combinatorial proof of the generalization

The original Beck conjecture, now a theorem due to Andrews, states that the difference in the number of parts in all partitions into odd parts and the number of parts in all strict partitions is equal to the number of partitions whose set of even parts has one element, and also to the number of partitions with exactly one part repeated. This is a companion identity to Euler's identity. The theorem has been generalized by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity for Franklin's identity and prove it via a modular refinement. We provide both analytical and combinatorial proofs. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also give a generalization to Franklin's identity of the second Beck-type companion identity proved by Andrews and Yang in their respective work.

We introduce new convolutions for the number of divisors function. We also provide combinatorial interpretations for some of the convolutions. In addition, we prove arithmetic properties for several restricted partitions functions used in the convolutions.

In this article, we explore the parity of sums of partition numbers at certain places in arithmetic progressions. In particular, we investigate pairs $$(a,b)\in \mathbb {N}^2$$ for which if and only if $$bn+1$$ is a square.

In the 1995 paper entitled “Noncommutative symmetric functions”, Gelfand et al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In contrast to the symmetric power sums, the quasisymmetric power sums have a more complex combinatorial description. This paper offers a first detailed exploration of these two relatively unstudied quasisymmetric bases, in which we show that they refine the classical symmetric power sum basis, we give transition matrices to other well-understood bases, and we provide explicit formulas for products of quasisymmetric power sums.

We consider the function $$r_s(n)$$ which gives the number of ways to write n as the sum of s squares. Since the generating functions for $$r_4(n)$$ and $$r_8(n)$$ are Lambert series, we use Merca’s factorization theorem for Lambert series to establish relationships between these functions and partitions into distinct parts. We also obtain convolutions involving overpartition functions as well as pentagonal recurrence formulas for $$r_4(n)$$ and $$r_8(n)$$ . These results lead to new connections between divisors and partitions.

In this paper, we give asymptotic formulas that combine the Euler–Riemann zeta function and the Chebyshev–Stirling numbers of the first kind. These results allow us to prove an asymptotic formula related to the nth complete homogeneous symmetric function, which was recently conjectured by the second author: $$\begin{aligned} h_{n}\left( 1,\left( \frac{k}{k+1}\right) ^2 ,\left( \frac{k}{k+2} \right) ^2 ,\ldots \right) \sim \left( {\begin{array}{c}2k\\ k\end{array}}\right) \quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$ A direct proof of this asymptotic formula, due to Gergő Nemes, is provided in Appendix.

The least $r$-gap, $g_r(\lambda)$, of a partition $\lambda$ is the smallest part of $\lambda$ appearing less than $r$ times. In this article we introduce two new partition functions involving least $r$-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler's partition function $p(n)$, polygonal numbers, and the new partition functions. To prove the results we use an interplay of combinatorial and $q$-series methods. We also give a combinatorial interpretation for $$\sum_{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2).$$, Comment: 10 pages

We use the representation theory of the quasisplit form G of SU(3) over a p-adic field to investigate whether certain quotients of the Bruhat--Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with G (which is a biregular bigraph) is Ramanujan if and only if G satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of PGL(2) considered by Lubotzky-Phillips-Sarnak. As a consequence, the classification by Rogawski of the automorphic spectrum of U(3) implies the existence of certain infinite families of Ramanujan bigraphs., 16 pages

A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products for which colouring pre-images of annuli centred at the origin allow us to describe the mappings with a high degree of fidelity. Additional graphics and animations are provided on the website of the project (http://math.holycross.edu/~cballant/complex/complex-functions.html).

We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive formula for this determinant in the case in which $P$ is a boolean algebra., Comment: 14 pages, AMS-TeX

We describe a representation of the local Hecke algebra for GLn in which the Hecke operators act on the vertices of the Bruhat-Tits building for SLn(Qp). We also give a geometric interpretation of this representation, characterizing the action of our operators on a vertex in terms of the endpoints of minimal walks in the building. This generalizes work of Serre who dened Hecke operators acting on the vertices of a tree (the building for SL2(Qp)).

We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat–Tits building of an inner form of \(\mathrm{SU}_{3}(\mathbb{Q}_{p})\). To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

LetFbe a totally real number field and let GLnbe the general linear group of rank n overF. Let р be a prime ideal ofFand Fрthe completion ofFwith respect to the valuation induced by р. We will consider a finite quotient of the affine building of the group GLnover the field Fр. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

We will construct a finite union of finite quotients of the affine building of the group GL3 over the field of p-adic numbers p. We will view this object as a hypergraph and estimate the spectrum of its underlying graph.

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors., Comment: 15 pages, improved argument that sequence of upper bounds in Theorem 5 is decreasing

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by \lambda^c its complement in a square partition (m^m). We conjecture a Schur-positivity criterion for symmetric functions of the form s_{\mu'}s_{\mu^c}-s_{\lambda'}s_{\lambda^c}, where \lambda is a partition of weight |\mu|-1 contained in \mu and the complement of \mu is taken in the same square partition as the complement of \lambda. We prove the conjecture in many cases., Comment: 28 pages, 16 figures

Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function $s_{\m}$ in the decomposition of an even power of the Vandermonde determinant in $n + 1$ variables in terms of the coefficient of the Schur function $s_{\l}$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $\m$ is obtained from the Young diagram of $\l$ by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11, Comment: 23 pages; extended abstract appeared in the Proceedings of FPSAC11

(2002). A Simple Proof of Rolle's Theorem for Finite Fields. The American Mathematical Monthly: Vol. 109, No. 1, pp. 72-74.

We use recent work of Jonah Blasiak (2012 arXiv:1209.2018) to prove a stability result for the coefficients in the Kronecker product of two Schur functions: one indexed by a hook partition and one indexed by a rectangle partition. We also give nearly sharp bounds for the size of the partition starting with which the Kronecker coefficients are stable. Moreover, we show that once the bound is reached, no new Schur functions appear in the decomposition of Kronecker product. We call this property superstability. Thus, one can recover the Schur decomposition of the Kronecker product from the smallest case in which the superstability holds. The bound for superstability is sharp. Our study of this particular case of the Kronecker product is motivated by its usefulness for the understanding of the quantum Hall effect (Scharf T et al 1994 J. Phys. A: Math. Gen 27 4211–9).

The Kronecker product of two Schur functions $s_{\lambda}$ and $s_{\mu}$, denoted $s_{\lambda}\ast s_{\mu}$, is defined as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of $n$, $\lambda$ and $\mu$, respectively. The coefficient, $g_{\lambda,\mu,\nu}$, of $s_{\nu}$ in $s_{\lambda}\ast s_{\mu}$ is equal to the multiplicity of the irreducible representation indexed by $\nu$ in the tensor product. In this paper we give an algorithm for expanding the Kronecker product $s_{(n-p,p)}\ast s_{\lambda}$ if $\lambda_1-\lambda_2\geq 2p$. As a consequence of this algorithm we obtain a formula for $g_{(n-p,p), \lambda ,\nu}$ in terms of the Littlewood-Richardson coefficients which does not involve cancellations. Another consequence of our algorithm is that if $\lambda_1-\lambda_2\geq 2p$ then every Kronecker coefficient in $s_{(n-p,p)}\ast s_{\lambda}$ is independent of $n$, in other words, $g_{(n-p,p),\lambda,\nu}$ is stable for all $\nu$.