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.

scholarly journals On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

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 A Short Approach to Catalan Numbers Modulo $2^r$

10.37236/664 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guoce Xin ◽  
Jing-Feng Xu
Keyword(s):  
Catalan Numbers ◽  
Binary Expansion ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Recent Classification ◽  
Motzkin Paths

We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.

scholarly journals On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

10.37236/2313 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Kendra Killpatrick
Keyword(s):  
Equivalence Classes ◽  
Catalan Numbers ◽  
Major Index ◽  
Powers Of 2 ◽  
Set Of Permutations ◽  
Adic Valuation ◽  
Permutation Statistic ◽  
Wilf Equivalence

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan.  In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$.  The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$.  We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result.  Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index.    In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$.  We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$.  In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

scholarly journals A new approach to the r-Whitney numbers by using combinatorial differential calculus

2019 ◽  
Vol 11 (2) ◽  
pp. 387-418
Author(s):  
Miguel A. Méndez ◽  
José L. Ramírez
Keyword(s):  
Differential Calculus ◽  
Differential Operators ◽  
Family Study ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Combinatorial Interpretation ◽  
New Approach ◽  
Whitney Numbers ◽  
Combinatorial Interpretations ◽  
Binomial Type

Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

scholarly journals Moments of Catalan Triangle Numbers

2020 ◽  
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Generating Functions ◽  
Catalan Numbers ◽  
Integer Sequences ◽  
Combinatorial Interpretations ◽  
Sums Of Powers ◽  
Alternating Sums

In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: ∑ k = 1 n k m B n , k j , ∑ k = 1 n + 1 2 k − 1 m A n , k j , for j , n ∈ N and m ∈ N ∪ 0 . We present their closed expressions for some values of m and j . Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

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 Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers

2020 ◽  
Vol 24 (3) ◽  
pp. 503-530
Author(s):  
Curtis Bennett ◽  
Juan Carrillo ◽  
John Machacek ◽  
Bruce E. Sagan
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Combinatorial Interpretations

scholarly journals The Delta Conjecture

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
James Haglund ◽  
Jeffrey B. Remmel ◽  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Function ◽  
Ordered Set ◽  
Macdonald Polynomials ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Combinatorial Interpretations ◽  
Llt Polynomials ◽  
Ordered Set Partitions ◽  
International Audience ◽  
Tesler Matrices

International audience We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

scholarly journals p, q-Stirling numbers and set partition statistics

1991 ◽  
Vol 56 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Michelle Wachs ◽  
Dennis White
Keyword(s):  
Stirling Numbers ◽  
Set Partition ◽  
Partition Statistics
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