For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not divisible by $l$ and occurring less than $k$ times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case $l=2$, we discuss the possibility of constructing a Schur-type companion of the Little Glaisher theorem via the weighted words.
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus of the present article is the analysis of combinatorial and arithmetical properties of them. The results include several combinatorial identities and recurrences as well as some properties of their $p$-adic valuations.
Extending the notion of $r$-(class) regular partitions, we define $(r_{1},...,r_{m})$-class regular partitions. A partition identity is presented and described by making use of the Glaisher correspondence.
Sometimes we need the approximate value of the partition number in a simple and efficient way. There are already several formulae to calculate the partition number p(n). But they are either inconvenient for most people (not majored in math) who do not want do write programs, or unsatisfying in accuracy. By bringing in two parameters in the Hardy-Ramanujan's Asymptotic formula and fitting the data of the two parameters by least square method, iteration method and some other special designed methods, several revised elementary estimation formulae with high accuracy for p(n) are obtained. With th…
The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$ and $s$. Classical notions such as Dyck paths and parking functions are naturally generalized by considering the set of partitions included in a given triangular partition. One of our striking results is that the restriction of the Young lattice to triangular partition has a planar Hasse diagram, wi…
We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.
Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations
In 1997, Bousquet-Mélou and Eriksson stated a broad generalization of Euler's distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvester's bijection. Subsequently, as an open questio…
Integer compositions, integer partitions, Fibonacci numbers, and generalizations of these have recently been shown to be interconnected via two-toned tilings of horizontal grids. In this article, we present refinements of two-toned tilings, describe functions which analyze them, and apply these to generalizations of integer compositions and partitions which interpolate between the two.
We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions. We provide closed forms for the generating functions of these…
Recently, Andrews and Paule studied Schmidt type partitions using MacMahon's Partition Analysis and obtained various interesting results. In this paper, we focus on the combinatorics of Schmidt type partition theorems and characterize them in a general and refined form. In addition, we also present some overpartition analogues of Schmidt type partition theorems.
We study oscillations in the eigenfunctions for a fractional Schrödinger operator on the real line. An argument in the spirit of Courant's nodal domain theorem applies to an associated local problem in the upper half plane and provides a bound on the number of nodal domains for the extensions of the eigenfunctions. Using the combinatorial properties of noncrossing partitions, we turn the nodal domain bound into an estimate for the number of sign changes in the eigenfunctions. We discuss applications in the periodic setting and the Steklov problem on planar domains.
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation.
We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In particular, we examine the correspondence between the wavefunctions of the XXZ type, Felderhof type and the boson type integrable models and symmetric polynomials such as the Schur, Grothendieck and symplectic Schur functions. We also give a brief report of our work on generalizing the correspondence between the Felderhof models and factorial Schur and symplectic …
In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.
We present a study of three families of Kronecker coefficients, which we describe in terms of reduced Kronecker coefficients. This study is grounded on the generating function of the coefficients, proved by a bijection between two combinatorial objects. This study includes the connection between plane partitions and these three families of reduced Kronecker coefficients, providing us their combinatorial interpretation. As an application, we verify that the saturation hypothesis holds for our three families of reduced Kronecker coefficients. The study also includes other interpretation in terms…
The partition number $π(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojević, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.
UPGMA（Unweighted Pair Group Method with Arithmetic Mean）等基于距离的方法继续在系统发育研究中发挥重要作用。我们使用多面体组合分析的自然细分的正正交诱导分类的输入向量根据树的拓扑结构返回的算法。划分格通知UPGMA树的研究。给出了n个分类单元上UPGMA锥的极值射线的一个封闭形式，并计算了n个分类单元上UPGMA锥的归一化体积。关键词：系统发育树，多面体组合，划分格
We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to Donaldson-Thomas and Pandharipande-Thomas theory which will be the subject of a forthcoming paper. The proo…
In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In a recent paper, the author generalised the first of these theorems to overpartitions, using a new technique which consists in going back and forth between $q$-difference equations on generating functions and recurrence equations on thei…
A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes using independent random variables. The principle outlines conditions under which the total variation distance between the true joint distribution and the approximation should be small, and was successfully exploited by Pittel in the cases of integer partitions and set partitions. We provide sufficient conditions for this principle to be true in a general context, valid for certain discrete probability di…
The central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series $f$ in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call $\mathcal{U}$ and $\mathcal{D}$ operators, whose action on the Schur sym…
In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of Glaisher's bijection, the map may be widely useful in other partition combinatorics. A previous conjecture concerning iterations of Glaisher's bijection is given a counterexample.
For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of param…
