An ordered partition of [n]:={1,2,..., n} is a sequence of its disjoint subsets whose union is [n]. The number of ordered partitions of [n] with k blocks is k!S(n,k), where S(n,k) is the Stirling number of second kind. In this paper we prove some refinements of this formula by showing that the generating function of some statistics on the set of ordered partitions of [n] with k blocks is a natural $q$-analogue of k!S(n,k). In particular, we prove several conjectures of Steingr\'ımsson. To this end, we construct a mapping from ordered partitions to walks in some digraphs and then, thanks to transfer-matrix method, we determine the corresponding generating functions by determinantal computations.
For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.
We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations.
A set partition of $[n] := \{1, 2, \dots, n \}$ is called {\em $r$-Stirling} if the numbers $1, 2, \dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k \leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r \leq k \leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of t…
The Stirling number of a simple graph is the number of partitions of its vertex set into a specific number of non-empty independent sets. In 2015, Engbers et al. showed that the coefficients in the normal ordering of a word $w$ in the alphabet $\{x,D\}$ subject to the relation $Dx=xD+1$ are equal to the Stirling number of certain graphs constructed from $w$. In this paper, we introduce graphical versions of the Stirling numbers of the first kind and the Lah numbers and show how they occur as coefficients in other normal ordering settings. Identities involving their $q$-analogues are also obtai…
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
Using Reiner's definition of Stirling numbers of type B of the second kind, we provide a 'balls into urns' approach for proving a generalization of a well-known identity concerning the classical Stirling numbers of the second kind: $x^n=\sum\limits_{k=0}^n{S(n,k)[x]_k}.$
The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls labeled $1$, $b_2$ balls labeled $2$, $\ldots$, $b_n$ balls labeled $n$ and $c_1+c_2+\ldots+c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled $2$, $\ldots$, $c_k$ cells labeled $k$ and then we called the number of ways to partition the set of these balls into non-empty cells of these types as the \textit{mixed partition numbers}. As an application, we g…
We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the $r$-Stirling numbers. We also introduce the associated $(S,r)$-Bell and $(S,r)$-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For $S$ with some extra structure, we show that the inverse of the $(S,r)$-Stirling matrix encodes th…
We define combinatorially a partial order on the set partitions and show that it is equivalent to the Bruhat-Chevalley-Renner order on the upper triangular matrices. By considering subposets consisting of set partitions with a fixed number of blocks, we introduce and investigate "Stirling posets." As we show, the Stirling posets have a hierarchy and they glue together to give the whole set partition poset. Moreover, we show that they (Stirling posets) are graded and EL-shellable. We offer various reformulations of their length functions and determine the recurrences for their length generating…
In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of weighted lattice path and tilings. We also present some new convolutions involving the complete and elementary symmetric functions. Additionally, we introduce different families of set partitions to give combinatorial interpretations for the modular $s$-Stirling numbers.
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.
We introduce and study the lattice of generalized partitions, called weighted partitions. This lattice possesses similar properties of the lattice of partitions. By use of the pictorial representation of a weighted partition, the total number is given by the successive Stirling transforms of the Stirling number of the second kind. We construct an explicit $EL$-labeling on the lattice, which implies this lattice is $EL$-shellable and hence shellable. We compute the Möbius function and the characteristic polynomial by use of a pictorial representation of a maximal decreasing chain. Further, a ma…
It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain partitioning process that gives rise to a sequence of sets of "nested" partitions. We prove that at stage m, the cardinality of the resulting set will equal the m-th order Bell number. This set-theoretic interpretation enables us to make a natural definition of higher order Stirling numbers and to study the combinatorics of these entities. The cardinality of the ele…
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling …
This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using $(l,r)$-Stirling of the first kind.
The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingrímsson over the set of permutations that avoid vincular pattern $1\underline{32}$, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related $q$-Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturall…
Stirling numbers, which count partitions of a set and permutations in the symmetric group, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of t…
We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the q-Stirling numbers of the second kind. Some of the statistics are generalizations of known statistics on set partitions, but others are entirely new. All the new ones are sums of two statistics, inspired by statistics on permutations, where one of the two statistics is based on a certain partial ordering of the blocks of a partition.
Broder introduced the r-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively in distinct subsets. Kim-Kim-Lee-Park constructed the degenerate r-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate r-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal orde…
On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families
Department of Computer science, University of Torbat-e Jam（托尔巴特-贾姆大学计算机系）；Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad（马什哈德费尔多西大学理学院纯数学系）
We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$, where the parts have a specific number of colors. As a consequence, some new relations between partitions, Bell numbers and Stirling number of the second kind are derived. We also derive a recursive formula for the number of unordered factorizations with $k$ different parts and a simple recursive formula for the number of partitions with $k$ different parts.
Given $R \subseteq \mathbb{N}$ let ${n \brace k}_R$, ${n \brack k}_R$, and $L(n,k)_R$ be the number of ways of partitioning the set $[n]$ into $k$ non-empty subsets, cycles and lists, respectively, with each block having cardinality in $R$. We refer to these as the $R$-restricted Stirling numbers of the second and first kind and the $R$-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind and first kind, and Lah numbers are ${n \brace k} = {n \brace k}_{\mathbb{N}}$, ${n \brack k} = {n \brack k}_{\mathbb{N}} $ and $L(n,k) = L(n,k)_{\mathbb{N}}$, res…
An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(σ)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $σ$ and obtained the formula for $|\mathcal{OP}_{n,k}(σ)|$ when the pattern $σ$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|\mathcal{OP}_{n,k}(σ)|$ when the pattern $σ$ is of length $3$. In this paper, we define a new patt…
In this paper, we study merging-free partitions with their canonical forms and run-sorted permutations. We give a combinatorial proof of the conjecture made by Nabawanda et al. We describe the distribution of the statistics of runs and right-to-left minima over the set of run-sorted permutations and we give the exponential generating function for their joint distribution. We show the number of right-to-left minima is given by the shifted distribution of the Stirling number of the second kind. We also prove that the non-crossing merging-free partitions are enumerated by powers of 2. We use one …
We construct an action of the Hecke algebra $H_n(q)$ on a quotient of the polynomial ring $F[x_1, \dots, x_n]$, where $F = \mathbb{Q}(q)$. The dimension of our quotient ring is the number of $k$-block ordered set partitions of $\{1, 2, \dots, n \}$. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at $q = 1$ and work of Huang-Rhoades at $q = 0$.
In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For this new combinatorial sequence we derive its exponential generating function, some recurrence relations, and combinatorial identities. We prove and present results using combinatorial arguments, generating functions, the symbolic method and Riordan arrays. For some specific cases we provide a combinatorial interpretation for the inverse matrix of the generaliz…
