scholarly journals Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets

10.37236/1264 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Partially Ordered Sets ◽  
Schur Function ◽  
Ordered Sets ◽  
Graded Poset ◽  
Kostka Polynomials ◽  
Relative Version ◽  
Definition Of ◽  
Formal Power

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power $q$, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

scholarly journals On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

10.37236/6732 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Gamma Delta ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
Leading Term ◽  
Definition Of

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals A Macdonald Vertex Operator and Standard Tableaux Statistics

10.37236/1383 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mike Zabrocki
Keyword(s):  
Vertex Operator ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Function ◽  
Inductive Proof ◽  
Integer Coefficients ◽  
The Family ◽  
The Relationship ◽  
Two Parameter

The two parameter family of coefficients $K_{\lambda \mu}(q,t)$ introduced by Macdonald are conjectured to $(q,t)$ count the standard tableaux of shape $\lambda $. If this conjecture is correct, then there exist statistics $a_\mu(T)$ and $b_\mu(T)$ such that the family of symmetric functions $H_\mu[X;q,t] = \sum_\lambda K_{\lambda \mu}(q,t) s_\lambda [X]$ are generating functions for the standard tableaux of size $|\mu|$ in the sense that $H_\mu[X;q,t] = \sum_{T} q^{a_\mu(T)} t^{b_\mu(T)} s_{\lambda (T)}[X]$ where the sum is over standard tableau of of size $|\mu|$. We give a formula for a symmetric function operator $H_2^{qt}$ with the property that $H_2^{qt} H_{(2^a1^b)}[X;q,t]= H_{(2^{a+1}1^b)}[X;q,t]$. This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that $H_{(2^a1^b)}[X;q,t]$ is the generating function for standard tableaux of size $2a+b$ (and hence that $K_{\lambda (2^a1^b)}(q,t)$ is a polynomial with non-negative integer coefficients). The inductive proof gives an algorithm for 'building' the standard tableaux of size $n+2$ from the standard tableaux of size $n$ and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics $a_\mu(T)$ and $b_\mu(T)$ when $\mu$ is of the form $(2^a1^b)$ and that these statistics prove conjectures about the relationship between adjacent rows of the $(q,t)$-Kostka matrix that were suggested by Lynne Butler.

scholarly journals The Murnaghan―Nakayama rule for k-Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Anne Schilling ◽  
Mike Zabrocki
Keyword(s):  
Explicit Formula ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Schur Function ◽  
Noncommutative Symmetric Functions ◽  
Power Sum ◽  
International Audience

International audience We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene. Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

scholarly journals On the expression of a symmetric function in terms of the elementary symmetric functions

1888 ◽  
Vol 7 ◽  
pp. 41-42
Author(s):  
R. E. Allardice
Keyword(s):  
Rational Function ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Proof ◽  
Elementary Symmetric Functions ◽  
Definition Of

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.

scholarly journals On the Schur Function Expansion of a Symmetric Quasi-symmetric Function

10.37236/8163 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Ira M. Gessel
Keyword(s):  
Simple Proof ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Function ◽  
Function Expansion

Egge, Loehr, and Warrington proved a formula for the Schur function expansion of a symmetric function in terms of its expansion in fundamental quasi-symmetric functions. Their formula involves the coefficients of a modified inverse Kostka matrix. Recently Garsia and Remmel gave a simpler reformulation of Egge, Loehr, and Warrington's result, with a new proof. We give here a simple proof of Garsia and Remmel's version, using a sign-reversing involution.

scholarly journals Place-Difference-Value Patterns: A Generalization of Generalized Permutation and Word Patterns

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Partially Ordered Sets ◽  
General Definition ◽  
Permutation Patterns ◽  
Ordered Sets ◽  
Combinatorial Objects ◽  
Permutation Pattern ◽  
New Type ◽  
Number Of Publications ◽  
Definition Of ◽  
Patterns In Permutations

AbstractMotivated by the study of Mahonian statistics, in 2000, Babson and Steingrímsson [Sém. Lothar. Comb] introduced the notion of a “generalized permutation pattern” (GP) which generalizes the concept of “classical” permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrímsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Mélou et al. [J. Comb. Theory A] introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets.In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

scholarly journals Hall-Littlewood Polynomials in terms of Yamanouchi words

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
International Audience ◽  
Leading Term ◽  
Definition Of

International audience This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

To the spectral theory of partially ordered sets

2018 ◽  
Vol 60 (3) ◽  
pp. 578-598
Author(s):  
Yu. L. Ershov ◽  
M. V. Schwidefsky
Keyword(s):  
Spectral Theory ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

Modelling Nondeterministic Concurrent Processes with Event Structures

1991 ◽  
Vol 14 (1) ◽  
pp. 39-73
Author(s):  
Rita Loogen ◽  
Ursula Goltz
Keyword(s):  
Dynamic Behaviour ◽  
Partially Ordered Sets ◽  
Parallel Composition ◽  
Ordered Sets ◽  
Communicating Sequential Processes ◽  
Partially Ordered ◽  
Event Structures ◽  
Transition Relation ◽  
Concurrent Processes ◽  
Sequential Processes

We present a non-interleaving model for non deterministic concurrent processes that is based on labelled event structures. We define operators on labelled event structures like parallel composition, nondeterministic combination, choice, prefixing and hiding. These operators correspond to the operations of the “Theory of Communicating Sequential Processes” (TCSP). Infinite processes are defined using the metric approach. The dynamic behaviour of event structures is defined by a transition relation which describes the execution of partially ordered sets of actions, abstracting from internal events.

Estimating the area under a receiver operating characteristic curve using partially ordered sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ehsan Zamanzade ◽  
Xinlei Wang
Keyword(s):  
Roc Curve ◽  
Characteristic Curve ◽  
Partially Ordered Sets ◽  
Sampling Technique ◽  
Real Data ◽  
Cost Effective ◽  
Ordered Sets ◽  
Operating Characteristics ◽  
Partially Ordered ◽  
Receiver Operating

AbstractRanked set sampling (RSS), known as a cost-effective sampling technique, requires that the ranker gives a complete ranking of the units in each set. Frey (2012) proposed a modification of RSS based on partially ordered sets, referred to as RSS-t in this paper, to allow the ranker to declare ties as much as he/she wishes. We consider the problem of estimating the area under a receiver operating characteristics (ROC) curve using RSS-t samples. The area under the ROC curve (AUC) is commonly used as a measure for the effectiveness of diagnostic markers. We develop six nonparametric estimators of the AUC with/without utilizing tie information based on different approaches. We then compare the estimators using a Monte Carlo simulation and an empirical study with real data from the National Health and Nutrition Examination Survey. The results show that utilizing tie information increases the efficiency of estimating the AUC. Suggestions about when to choose which estimator are also made available to practitioners.

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