scholarly journals Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions

10.37236/2131 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Pietro Mongelli
Keyword(s):  
Spectral Theory ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial  interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.

scholarly journals 2-Adic valuations of Stirling numbers of the first kind

2019 ◽  
Vol 15 (09) ◽  
pp. 1827-1855 ◽  
Author(s):  
Min Qiu ◽  
Shaofang Hong
Keyword(s):  
Explicit Formula ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Disjoint Cycles ◽  
Adic Valuation ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be positive integers. We denote by [Formula: see text] the 2-adic valuation of [Formula: see text]. The Stirling numbers of the first kind, denoted by [Formula: see text], count the number of permutations of [Formula: see text] elements with [Formula: see text] disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the [Formula: see text]-adic valuations of [Formula: see text]. In this paper, by introducing the concept of [Formula: see text]th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of [Formula: see text]. We also prove that [Formula: see text] holds for all integers [Formula: see text] between 1 and [Formula: see text]. As a corollary, we show that [Formula: see text] if [Formula: see text] is odd and [Formula: see text]. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if [Formula: see text], then [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the [Formula: see text]th elementary symmetric functions of [Formula: see text]. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

scholarly journals Combinatorial Interpretations of the Jacobi-Stirling Numbers

10.37236/342 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yoann Gelineau ◽  
Jiang Zeng
Keyword(s):  
Spectral Theory ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations ◽  
Central Factorial Numbers

The Jacobi-Stirling numbers of the first and second kinds were introduced in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.

scholarly journals Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

10.37236/1044 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Andrius Kulikauskas ◽  
Jeffrey Remmel
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Transition Matrices ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations ◽  
Lyndon Words

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

scholarly journals A note on extrema of linear combinations of elementary symmetric functions

2012 ◽  
Vol 60 (2) ◽  
pp. 219-224 ◽  
Author(s):  
Alexander Kovačec ◽  
Salma Kuhlmann ◽  
Cordian Riener
Keyword(s):  
Symmetric Functions ◽  
Linear Combinations ◽  
Elementary Symmetric Functions

scholarly journals Some New Methods in the Theory of $m$-Quasi-Invariants

10.37236/1877 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
J. Bell ◽  
A. M. Garsia ◽  
N. Wallach
Keyword(s):  
Symmetric Functions ◽  
Free Module ◽  
Finitely Generated ◽  
Graded Algebras ◽  
New Approach ◽  
Elementary Symmetric Functions ◽  
New Methods ◽  
Regular Sequences

We introduce here a new approach to the study of $m$-quasi-invariants. This approach consists in representing $m$-quasi-invariants as $N^{tuples}$ of invariants. Then conditions are sought which characterize such $N^{tuples}$. We study here the case of $S_3$ $m$-quasi-invariants. This leads to an interesting free module of triplets of polynomials in the elementary symmetric functions $e_1,e_2,e_3$ which explains certain observed properties of $S_3$ $m$-quasi-invariants. We also use basic results on finitely generated graded algebras to derive some general facts about regular sequences of $S_n$ $m$-quasi-invariants

scholarly journals A Unified View of Determinantal Expansions for Jack Polynomials

10.37236/1547 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Leigh Roberts
Keyword(s):  
Differential Geometry ◽  
Symmetric Functions ◽  
Jack Polynomials ◽  
Elementary Symmetric Functions ◽  
Fundamental Symmetry ◽  
Unified View

Recently Lapointe et. al. [3] have expressed Jack Polynomials as determinants in monomial symmetric functions $m_\lambda$. We express these polynomials as determinants in elementary symmetric functions $e_\lambda$, showing a fundamental symmetry between these two expansions. Moreover, both expansions are obtained indifferently by applying the Calogero-Sutherland operator in physics or quasi Laplace Beltrami operators arising from differential geometry and statistics. Examples are given, and comments on the sparseness of the determinants so obtained conclude the paper.

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

Sign in / Sign up

Export Citation Format

Share Document