Ascent Sequences and Upper Triangular Matrices Containing Non-Negative Integers

This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics on these structures under this bijection and prove that some of these statistics are equidistributed. Several special classes of matrices are shown to have simple formulations in terms of ascent sequences. Binary matrices are shown to correspond to ascent sequences with no two adjacent entries the same. Bidiagonal matrices are shown to be related to order-consecutive set partitions and a simple condition on the ascent sequences generate this class.

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Composition Matrices, (2+2)-Free Posets and their Specializations

The Electronic Journal of Combinatorics ◽

10.37236/531 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 7

Author(s):

Mark Dukes ◽

Vít Jelínek ◽

Martina Kubitzke

Keyword(s):

Simple Condition ◽

Triangular Matrices ◽

Upper Triangular Matrices

In this paper we present a bijection between composition matrices and ($\mathbf{2+2}$)-free posets. This bijection maps partition matrices to factorial posets, and induces a bijection from upper triangular matrices with non-negative entries having no rows or columns of zeros to unlabeled ($\mathbf{2+2}$)-free posets. Chains in a ($\mathbf{2+2}$)-free poset are shown to correspond to entries in the associated composition matrix whose hooks satisfy a simple condition. It is shown that the action of taking the dual of a poset corresponds to reflecting the associated composition matrix in its anti-diagonal. We further characterize posets which are both ($\mathbf{2+2}$)- and ($\mathbf{3+1}$)-free by certain properties of their associated composition matrices.

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Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

The Electronic Journal of Combinatorics ◽

10.37236/8241 ◽

2019 ◽

Vol 26 (2) ◽

Author(s):

Megan Ly

Keyword(s):

Irreducible Representations ◽

Character Theory ◽

Set Partition ◽

Set Partitions ◽

Triangular Matrices ◽

Upper Triangular Matrices ◽

Weyl Duality ◽

Supercharacter Theory ◽

Combinatorial Representation Theory ◽

Centralizer Algebra

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph.

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Counting self-dual interval orders

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2932 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Vít Jelínek

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Interval Orders ◽

Maximal Elements ◽

Triangular Matrices ◽

Bijective Correspondence ◽

Upper Triangular Matrices ◽

Positive Entry ◽

International Audience ◽

Natural Statistics

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

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Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$ (extended abstract)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2840 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Stephen Lewis ◽

Nathaniel Thiem

Keyword(s):

Tensor Product ◽

Symmetric Group ◽

Nonnegative Integer ◽

Set Partitions ◽

Triangular Matrices ◽

Upper Triangular Matrices ◽

Branching Rules ◽

International Audience ◽

Supercharacter Theory ◽

Donne Lieu

International audience The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q) \subseteq U_n(q)$ for $m \leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs. La théorie standard des supercaractères des matrices triangulaires supérieures unipotentes finies $U_n(q)$ donne lieu à une merveilleuse combinatoire basée sur les partitions d'ensembles. Comme avec la théorie des représentations du groupe symétrique, Les plongements $U_m(q) \subseteq U_n(q)$ pour $m \leq n$ mènent aux règles de branchement. Diaconis et Isaacs ont montré que la restriction d'un supercaractère de $U_n(q)$ est une combinaison linéaire des supercaractères de $U_m(q)$ avec des coefficients entiers non négatifs (en fait, elle est polynomiale en $q$). Dans une première étape vers la compréhension de la combinatoire des coefficients dans les règles de branchement des supercaractères de $U_n(q)$, ce texte caractérise les coefficients non nuls dans la restriction d'un supercaractère et dans le produit des tenseurs de deux supercaractères. Ces conditions sont données uniformément en termes des couplages complets dans des graphes bipartis.

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Restricting Supercharacters of the Finite Group of Unipotent Uppertriangular Matrices

The Electronic Journal of Combinatorics ◽

10.37236/112 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 12

Author(s):

Nathaniel Thiem ◽

Vidya Venkateswaran

Keyword(s):

Finite Group ◽

Representation Theory ◽

Finite Field ◽

Symmetric Group ◽

Irreducible Representations ◽

Combinatorial Theory ◽

Set Partitions ◽

Triangular Matrices ◽

Upper Triangular Matrices ◽

Supercharacter Theory

It is well-known that understanding the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions).

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Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2698 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Nathaniel Thiem

Keyword(s):

Symmetric Functions ◽

Combinatorial Structure ◽

Classical Representation ◽

Set Partitions ◽

Triangular Matrices ◽

Upper Triangular Matrices ◽

Branching Rules ◽

International Audience ◽

Structure Constants ◽

Supercharacter Theory

International audience It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters.

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