scholarly journals The Rees product of the cubical lattice with the chain

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Patricia Muldoon ◽  
Margaret A. Readdy
Keyword(s):  
Möbius Function ◽  
Explicit Formulas ◽  
Bijective Proof ◽  
Mobius Function ◽  
International Audience

International audience We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof. Nous étudions des propriétés énumératives et homologiques du produit de Rees du treillis cubique avec la chaîne. Nous donnons plusieurs formules explicites de la fonction de Möbius de ce poset. La dernière de ces formules est exprimée en termes du permanent d’une matrice et le résultat est donné par une preuve bijective.

scholarly journals How to get the weak order out of a digraph ?

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Francois Viard
Keyword(s):  
Wreath Product ◽  
Symmetric Function ◽  
Coxeter Groups ◽  
Symmetric Functions ◽  
Weak Order ◽  
Möbius Function ◽  
Mobius Function ◽  
Acyclic Digraph ◽  
International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, ainsi qu’une variante de l’ordre faible sur les produits en couronne &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$<sub>$n-1$</sub> où l’on obtient les séries de Stanley classiques.

scholarly journals A Formula for the Möbius Function of the Permutation Poset Based on a Topological Decomposition

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jason P Smith
Keyword(s):  
Significant Proportion ◽  
Möbius Function ◽  
Permutation Patterns ◽  
Mobius Function ◽  
International Audience ◽  
Definition Of ◽  
Object Of Study ◽  
Pattern Containment ◽  
Exponential Complexity ◽  
Exponential Part

International audience The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.

scholarly journals The Möbius function of generalized subword order

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Peter R. W. McNamara ◽  
Bruce E. Sagan
Keyword(s):  
Partial Order ◽  
Finite Length ◽  
Simple Formula ◽  
Möbius Function ◽  
Finite Poset ◽  
Mobius Function ◽  
Nous Permet ◽  
International Audience ◽  
Uniform Manner ◽  
A Chain

International audience Let $P$ be a poset and let $P^*$ be the set of all finite length words over $P$. Generalized subword order is the partial order on $P^*$ obtained by letting $u≤ w$ if and only if there is a subword $u'$ of $w$ having the same length as $u$ such that each element of $u$ is less than or equal to the corresponding element of $u'$ in the partial order on $P$. Classical subword order arises when $P$ is an antichain, while letting $P$ be a chain gives an order on compositions. For any finite poset $P$, we give a simple formula for the Möbius function of $P^*$ in terms of the Möbius function of $P$. This permits us to rederive in an easy and uniform manner previous results of Björner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in $P^*$ for any finite $P$ of rank at most 1. Soit $P$ un ensemble partiellement ordonné et soit $P^*$ l'ensemble des mots de longueur finie sur $P$. On définit l'ordre des sous-mots généralisé comme l'ordre partiel sur $P^*$ obtenu en posant $u≤ w$ s'il existe un sous-mot $u'$ de $w$ ayant la même longueur que $u$, tel que chaque élément de $u$ soit plus petit ou égal à l'élément correspondant de $u'$ dans l'ordre partiel sur $P$. L'ordre des sous-mots classique correspond au cas où $P$ est une antichaîne ; tandis que si P est une chaîne, on obtient un ordre sur les compositions. Pour tout ensemble partiellement ordonné fini $P$, nous donnons une formule simple pour la fonction de Möbius de $P^*$ en fonction de celle de $P$. Cela nous permet de retrouver de manière simple et uniforme des résultats de Björner, Sagan et Vatter, et de Tomie. Nous sommes aussi en mesure de déterminer le type d'homotopie de tous les intervalles de $P^*$ pour n'importe quel $P$ fini de rang au plus 1.

scholarly journals Factorization of the Characteristic Polynomial

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Joshua Hallam ◽  
Bruce Sagan
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Finite Lattice ◽  
Möbius Function ◽  
New Method ◽  
Mobius Function ◽  
International Audience

International audience We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our method gives two simple conditions under which the characteristic polynomial factors. We will see that Stanley's Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate a new result in graph theory and give new proofs of some classic results concerning the Möbius function.

scholarly journals The Möbius function of separable permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Vít Jelínek ◽  
Eva Jelínková ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Computationally Efficient ◽  
Mobius Function ◽  
Nous Montrons ◽  
International Audience ◽  
Separable Permutations ◽  
Pattern Containment

International audience A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. Using the notion of separating tree, we give a computationally efficient formula for the Möbius function of an interval $(q,p)$ in the poset of separable permutations ordered by pattern containment. A consequence of the formula is that the Möbius function of such an interval $(q,p)$ is bounded by the number of occurrences of $q$ as a pattern in $p$. The formula also implies that for any separable permutation $p$ the Möbius function of $(1,p)$ is either 0, 1 or -1. Une permutation est séparable si elle peut être générée á partir de la permutation 1 par des sommes directes et des sommes indirectes, ou de façon équivalente, si elle évite les motifs 2413 et 3142. En utilisant le concept de l'arbre séparant, nous donnons une formule pour le calcul efficace de la fonction de Möbius d'un intervalle de $(q, p)$ dans l'ensemble partiellement ordonné des permutations séparables. Une conséquence est que la fonction de Möbius de $(q,p)$ pour $q$ et $p$ séparables est bornée par le nombre d'occurrences de $q$ comme un motif en $p$. Nous montrons aussi que pour une permutation $p$ séparable, la fonction de Möbius de $(1,p)$ est soit 0, 1 ou -1.

scholarly journals On an exponential sum involving the Möbius function

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
H. Maier ◽  
A Sankaranarayanan
Keyword(s):  
Upper Bound ◽  
Exponential Sum ◽  
Möbius Function ◽  
Absolute Value ◽  
Mobius Function ◽  
International Audience ◽  
The Absolute

International audience In this paper we study the upper bound for the absolute value of the exponential sum related to the Möbius function unconditionally and present some interesting applications also.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

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