scholarly journals The Complete cd-Index of Boolean Lattices

10.37236/5100 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Neil J.Y. Fan ◽  
Liao He
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Boolean Lattice ◽  
Combinatorial Interpretation ◽  
Euler Numbers ◽  
Combinatorial Interpretations ◽  
Bruhat Graph ◽  
Boolean Lattices ◽  
One To One ◽  
Entringer Numbers

Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.

scholarly journals Shortest path poset of finite Coxeter groups

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Saúl A. Blanco
Keyword(s):  
Shortest Path ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Shortest Paths ◽  
Combinatorial Interpretation ◽  
Nous Permet ◽  
Finite Coxeter Group ◽  
International Audience ◽  
Absolute Length ◽  
Bruhat Graph

International audience We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$. Nous définissons un poset en utilisant le plus court chemin entre l'identité et le plus long mot de $W, w_0$, dans le graph de Bruhat du groupe finie Coxeter, $W$. Nous prouvons que ce poset est l'union de posets Boolean du même rang que la longueur absolute de $w_0$; ça signifie que tous les plus courts chemins, étiquetés par réflexions $t_1,\ldots, t_m$ sont totalement commutatives. Ça nous permet de donner une interprétation combinatoire aux termes avec le moindre grade dans le $\textbf{cd}$-index complet de $W$.

scholarly journals Chains in generalized Boolean lattices

1976 ◽  
Vol 21 (2) ◽  
pp. 234-240
Author(s):  
Richard D. Byrd ◽  
Roberto A. Mena
Keyword(s):  
Distributive Lattice ◽  
Boolean Lattice ◽  
Boolean Lattices ◽  
A Chain

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

scholarly journals Extensions of Partial Cyclic Orders, Euler Numbers and Multidimensional Boustrophedons

10.37236/7145 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sanjay Ramassamy
Keyword(s):  
Cyclic Order ◽  
Euler Numbers ◽  
Entringer Numbers

We enumerate total cyclic orders on $\left\{x_1,\ldots,x_n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(x_i,x_{i+1},x_{i+2})$, with indices taken modulo $n$. In some cases, the problem reduces to the enumeration of descent classes of permutations, which is done via the boustrophedon construction. In other cases, we solve the question by introducing multidimensional versions of the boustrophedon. In particular we find new interpretations for the Euler up/down numbers and the Entringer numbers.

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 The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains

10.37236/9034 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Hong-Bin Chen ◽  
Yen-Jen Cheng ◽  
Wei-Tian Li ◽  
Chia-An Liu
Keyword(s):  
Lower Bounds ◽  
Boolean Lattice ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Ramsey Type ◽  
Boolean Lattices

Motivated by the paper, Boolean lattices: Ramsey properties and embeddings Order, 34 (2) (2017), of Axenovich and Walzer, we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring of elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$ as an induced subposet. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also derive some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.

scholarly journals MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing

2014 ◽  
Author(s):  
Daniel J. Greenhoe
Keyword(s):  
Boolean Lattice ◽  
Symbolic Sequence ◽  
Sequence Processing ◽  
Linear Subspaces ◽  
Genomic Signal Processing ◽  
Set Inclusion ◽  
Symbolic Sequences ◽  
Special Lattice ◽  
Boolean Lattices ◽  
Selection Of

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").

scholarly journals Representation of Boolean hypolattices

1981 ◽  
Vol 24 (3) ◽  
pp. 389-404 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Representation Theorem ◽  
Boolean Lattice ◽  
Principal Result ◽  
Direct Sum ◽  
Boolean Lattices ◽  
Stone Representation ◽  
Relative Convexity ◽  
Small Product

The principal result is a representation theorem for relatively-distributive, relatively complemented hypolattices with zero, generalizing the Stone representation theorem for a Boolean lattice. It uses the small product of a family of Boolean lattices which are maximal sublattices of the hypolattice. The paper also characterizes the maximal sublattices when the hypolattice is coherent; and it gives several examples of hypolattices, including hypolattices of subgroups and of ideals by direct sum, and examples from relative convexity, relative closure, and cofinality.

scholarly journals Combinatorial Interpretations of the Kreweras Triangle in Terms of Subset Tuples

10.37236/7531 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Combinatorial Interpretations ◽  
Dumont Permutations

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

scholarly journals Trees and Meta-Fibonacci Sequences

10.37236/218 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Abraham Isgur ◽  
David Reiss ◽  
Stephen Tanny
Keyword(s):  
Closed Form ◽  
Empirical Evidence ◽  
Nonnegative Integer ◽  
Asymptotic Estimates ◽  
Combinatorial Interpretation ◽  
Labeled Trees ◽  
Special Cases ◽  
Fibonacci Sequences

For $k>1$ and nonnegative integer parameters $a_p, b_p$, $p = 1..k$, we analyze the solutions to the meta-Fibonacci recursion $C(n)=\sum_{p=1}^k C(n-a_p-C(n-b_p))$, where the parameters $a_p, b_p$, $p = 1..k$ satisfy a specific constraint. For $k=2$ we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include the Conolly recursion and all of its generalizations that have been studied to date. We show that the solutions for all the recursions defined by the parameters in these families have a natural combinatorial interpretation: they count the number of labels on the leaves of certain infinite labeled trees, where the number of labels on each node in the tree is determined by the parameters. This combinatorial interpretation enables us to determine various new results concerning these sequences, including a closed form, and to derive asymptotic estimates. Our results broadly generalize and unify recent findings of this type relating to certain of these meta-Fibonacci sequences. At the same time they indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.

scholarly journals Path tableaux and combinatorial interpretations of immanants for class functions on $S_n$

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sam Clearman ◽  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Irreducible Character ◽  
Symmetric Function ◽  
Nonnegative Matrix ◽  
Schur Function ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations ◽  
Characteristic Map ◽  
International Audience ◽  
Totally Nonnegative Matrix

International audience Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$. Soit $χ ^λ$ le caractère irréductible de $S_n$ qui correspond à la partition λ de l'entier n, ou de manière équivalente, la préimage de la fonction de Schur $s_λ$ par l'application caractéristique de Frobenius. Soit $\phi ^λ$ la fonction $S_n →ℂ$ qui est la préimage de la fonction symétrique monomiale m_λ . La valeur du caractère irréductible immanent $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ est non négative pour chaque matrice totalement non négative. Nous donnons une interprétation combinatoire de la valeur $Imm_λ (A)$ lorsque $λ$ est une partition en équerre. Stembridge a conjecturé que la valeur de l'immanent monomial $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ de $\phi ^λ$ est elle aussi non négative pour chaque matrice totalement non négative. Nous confirmons cette conjecture quand λ satisfait $λ _1 ≤2$, et nous donnons une interprétation combinatoire de $Imm_{{\phi} ^λ} (A)$ dans ce cas.

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