scholarly journals Partitions, Rooks, and Symmetric Functions in Noncommuting Variables

10.37236/1999 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Mahir Bilen Can ◽  
Bruce E. Sagan
Keyword(s):  
Vector Space ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Set Partitions ◽  
Formal Vector ◽  
Rook Theory ◽  
Rook Placements

Let $\Pi_n$ denote the set of all set partitions of $\{1,2,\ldots,n\}$. We consider two subsets of $\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ${\cal E}_n\subseteq\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\cal T}_{n-1}$. Given $\pi\in\Pi_m$ and $\sigma\in\Pi_n$, define their slash product to be $\pi|\sigma=\pi\cup(\sigma+m)\in\Pi_{m+n}$ where $\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\sigma$. Call $\tau$ atomic if it can not be written as a nontrivial slash product and let ${\cal A}_n\subseteq\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, ${\cal E}_n={\cal A}_n$ for all $n\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks.

scholarly journals Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

2008 ◽  
Vol 60 (2) ◽  
pp. 266-296 ◽  
Author(s):  
Nantel Bergeron ◽  
Christophe Reutenauer ◽  
Mercedes Rosas ◽  
Mike Zabrocki
Keyword(s):  
Hopf Algebra ◽  
Symmetric Group ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Set Partitions ◽  
Noncommutative Symmetric Functions ◽  
Hopf Algebra Structure ◽  
Natural Inclusion ◽  
Noncommutative Polynomials ◽  
Noncommutative Setting

AbstractWe introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a natural inclusion of the Hopf algebra of noncommutative symmetric functions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials and conclude two analogues of Chevalley’s theorem in the noncommutative setting.

scholarly journals Forms with O-Orthogonal Lie Algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 125-126
Author(s):  
Frank Servedio
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Homogeneous Polynomial ◽  
Algebra Structure ◽  
Orthogonal Lie Algebra ◽  
Coordinate Functions

A form P of degree r is a homogeneous polynomial in k[Yi, …, Yn] on kn, k a field; Yi are the coordinate functions on kn. Let V(n, r) denote the k-vector space of forms of degree r. Mn(k) = Endk(kn) has canonical Lie algebra structure with [A, B] = AB-BA and it acts as a k-Lie Algebra of kderivations of degree 0 on k[Yi, …, Yn] defined by setting D(A)Y= Yo(-A) for A∈Endk(kn), Y∈V(n,l) = Homk(kn, k) and extending as a k-derivation. Define the orthogonal Lie Algebra, LO(P), of P by LO(P) =

scholarly journals Correlation Angles and Inner Products: Application to a Problem from Physics

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Adam Towsley ◽  
Jonathan Pakianathan ◽  
David H. Douglass
Keyword(s):  
Vector Space ◽  
Dimensional Space ◽  
Random Variables ◽  
Climate Science ◽  
Inner Product ◽  
Correlation Studies ◽  
Formal Vector ◽  
Inner Products

Covariance is used as an inner product on a formal vector space built on random variables to define measures of correlation across a set of vectors in a -dimensional space. For , one has the diameter; for , one has an area. These concepts are directly applied to correlation studies in climate science.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Bijections on m-level Rook Placements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Kenneth Barrese ◽  
Bruce Sagan
Keyword(s):  
Special Class ◽  
Symmetric Functions ◽  
Elementary Symmetric Functions ◽  
Rook Placements ◽  
Rook Numbers ◽  
Ferrers Board ◽  
International Audience ◽  
Ferrers Boards

International audience Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne. Nous considérons les rangs d’un échiquier partagés en ensembles de $m$ rangs appelés les niveaux. Un $m$-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de $m$-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de $m$-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne.

scholarly journals Rota–Baxter operators on a sum of fields

2019 ◽  
Vol 19 (06) ◽  
pp. 2050118
Author(s):  
V. Gubarev
Keyword(s):  
Vector Space ◽  
Algebra Structure ◽  
Direct Sum ◽  
Sum Formula

We count the number of all Rota–Baxter operators (RB-operators) on a finite direct sum [Formula: see text] of fields and count all of them up to conjugation with an automorphism. We also study RB-operators on [Formula: see text] corresponding to a decomposition of [Formula: see text] into a direct vector space sum of two subalgebras. We show that every algebra structure induced on [Formula: see text] by a RB-operator of nonzero weight is isomorphic to [Formula: see text].

A finite dimensional 𝐴∞ algebra example

2010 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
Michael P. Allocca ◽  
Tom Lada
Keyword(s):  
Vector Space ◽  
Algebra Structure ◽  
Finite Dimensional

Abstract We construct an example of an 𝐴∞ algebra structure defined over a finite dimensional graded vector space.

scholarly journals Tying up baric algebras

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Antonio Oller-Marcén
Keyword(s):  
Vector Space ◽  
Algebra Structure ◽  
Zero Characteristic ◽  
Characteristic Case ◽  
Baric Algebras ◽  
Baric Algebra

AbstractGiven two baric algebras (A 1, ω 1) and (A 2, ω 2) we describe a way to define a new baric algebra structure over the vector space A 1 ⊕ A 2, which we shall denote (A 1 ⋈ A 2, ω 1 ⋈ ω 2). We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form A 1 ⋈ A 2 in the associative, coutable-dimensional, zero-characteristic case are classified.

scholarly journals Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Adam Doliwa
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Natural Basis ◽  
Dual Algebra ◽  
Power Sum ◽  
Hopf Algebra Structure

We introduce a coloured generalization  $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions  described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables.  We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

scholarly journals Rook Theory, Generalized Stirling Numbers and $(p,q)$-Analogues

10.37236/1837 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
J. B. Remmel ◽  
Michelle L. Wachs
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Set Partitions ◽  
Growth Functions ◽  
Rook Theory ◽  
Signed Permutations ◽  
Generalized Stirling Numbers ◽  
Alpha Beta ◽  
Natural Statistics ◽  
Appl Math

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

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