Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].     

Analytic proof of the partition identity A 5,3,3(n)=B 5,3,3 0 (n)

In this paper we give an analytic proof of the identity A _5,3,3(n)=B _5,3,3⁰(n), where A _5,3,3(n) counts the number of partitions of n subject to certain restrictions on their parts, and B _5,3,3⁰(n) counts the number of partitions of n subject to certain other restrictions on their parts, both too long to be stated in the abstract. Our proof establishes actually a refinement of that partition identity. The original identity was first discovered by the first author jointly with M. Ruby Salestina and S.R. Sudarshan in [Proceedings of the International Conference on Analytic Number Theory with Special Emphasis on L-functions, Ramanujan Math. Soc., Mysore, 2005, pp. 57–70], where it was also given a combinatorial proof, thus answering a question of Andrews. Research partially supported by EC’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe.”     

Computing a pyramid partition generating function with dimer shuffling

We verify a recent conjecture of Kenyon/Szendr?i by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁x₂−x₃x₄=0}⊂C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp [Noam Elkies, Greg Kuperberg, Michael Larsen, James Propp, Alternating sign matrices and domino tilings. II, J. Algebraic Combin. 1 (3) (1992) 219-234].     

Set partitions and moments of random variables

It is well known that the sequence of Bell numbers (Bn)_n?0 (Bn being the number of partitions of the set [n]) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence (Bn+1)_n?0 is the sequence of moments of 1+τ. In this paper, we generalize these results by showing that both and (where is the number of m-partitions of [n], as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when is replaced by the number of ordered m-partitions of [n]. In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes.     

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

A brief history of partitions of numbers,partition functions and their modern applications

‘Number rules the universe.’

The Pythagoras     

Elementary proofs of infinitely many congruences for k-elongated partition diamonds

In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers $d_k(n)$ produced by summing the links of k-elongated plane partition diamonds of length n is given by $\frac{{(q^2;q^2)}_{\infty }^k}{{(q;q)}_{\infty }^{3k+1}}$ for each $k\ge 1$. A significant portion of their recent paper involves proving several congruence properties satisfied by $d_1,d_2$ and $d_3$, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions $d_k$ for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results.     

An elementary derivation of the asymptotics of partition functions

Let S_a,b = {an+b:n ≥ 0 } where n is an integer. Let P_a,b(n) denote the number of partitions of n into elements of S_a,b. In particular, we have the generating function,

A family of Bell transformations

We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full range of applications remains unexplored, in this paper we show a glimpse of the versatility of Bell transformations by discussing the enumeration of several combinatorial configurations, including rational Dyck paths, rooted planar maps, and certain classes of permutations.     

Resolution of a conjecture of Andrews and Lewis involving cranks of partitions

Andrews and Lewis have conjectured that the sign of the number of partitions of with crank congruent to 0 mod 3, minus the number of partitions of with crank congruent to 1 mod 3, is determined by the congruence class of mod 3 apart from a finite number of specific exceptions. We prove this by using the ``circle method" to approximate the value of this difference to great enough accuracy to determine its sign for all sufficiently large .

     

The perimeter of words

We define $\left[k\right]=\{1,2,\dots ,k\}$ to be a (totally ordered) alphabet on $k$ letters. A word $w$ of length $n$ on the alphabet $\left[k\right]$ is an element of ${\left[k\right]}^n$. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the $x$-axis) in which the height of the $i$th column equals the size of the $i$th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length $n$ over the alphabet $\left[k\right]$. We also show how the mean and variance can be obtained using a direct counting method.     

Sums of products of Cauchy numbers

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.     

On Cesàro means, Kaplan classes and a conjecture of S.P. Robinson

We establish special cases of a conjecture of S.P. Robinson [S.P. Robinson, Approximate identities for certain dual classes, DPhil thesis, University of York, UK, 1996] concerning Cesàro means of certain classes of analytic functions in the unit disk. This has applications, for instance, to the so-called Kaplan classes and subordination under ‘linearly accessible’ functions.     

On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum

     

Hypergraphs and a functional equation of Bouwkamp and de Bruijn

Let . Bouwkamp and de Bruijn found that there exists a power series Ψ(u,v) satisfying the equation . We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Φ(u,0) and Ψ(u,0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversion to count hypertrees by number of vertices and number of edges of a specified size.     

Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi

We study the asymptotic expansion of the first Dirichlet eigenvalue of certain families of triangles and of rhombi as a singular limit is approached. In certain cases, which include isosceles and right triangles, we obtain the exact value of all the coefficients of the unbounded terms in the asymptotic expansion as the angle opening approaches zero, plus the constant term and estimates on the remainder. For rhombi and other triangle families such as isosceles triangles where now the angle opening approaches π, we have the first two terms plus bounds on the remainder. These results are based on new upper and lower bounds for these domains whose asymptotic expansions coincide up to the orders mentioned. Apart from being accurate near the singular limits considered, our lower bounds for the rhombus improve upon the bound by Hooker and Protter for angles up to approximately 22° and in the range (31°,54°). These results also show that the asymptotic expansion around the degenerate case of the isosceles triangle with vanishing angle opening depends on the path used to approach it.     

On the length of the tail of a vector space partition

A vector space partition of a finite dimensional vector space V=V(n,q) of dimension n over a finite field with q elements, is a collection of subspaces U₁,U₂,…,U_t with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of consists of the subspaces of least dimension d₁ in , and the length n₁ of the tail is the number of subspaces in the tail. Let d₂ denote the second least dimension in .Two cases are considered: the integer q^d₂−d₁ does not divide respective divides n₁. In the first case it is proved that if 2d₁>d₂ then n₁≥q^d₁+1 and if 2d₁≤d₂ then either n₁=(q^d₂−1)/(q^d₁−1) or n₁>2q^d₂−d₁. These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d₁ respectively d₂.In case q^d₂−d₁ divides n₁ it is shown that if d₂<2d₁ then n₁≥q^d₂−q^d₁+q^d₂−d₁ and if 2d₁≤d₂ then n₁≥q^d₂. The last bound is also shown to be tight.The results considerably improve earlier found lower bounds on the length of the tail.     

Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contracted measure of zeros of regular, uniform, Wigner, Weyl, Karamata and hypergeometric types are explicitly identified. Some illustrative examples are given.