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Multiplicative structures of the immaculate basis of non-commutative symmetric functions

JOURNAL OF COMBINATORIAL THEORY SERIES A 2017年 152卷 10-44页

作者： Berg, Chris Bergeron, Nantel Saliola, Franco Serrano, Luis Zabrocki, Mike

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the basis of Schur functions for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Mumaghan-Nakayama rule. A surprising relation develops among non commutative Littlewood Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Structure constants Immaculate functions Quasisymmetric functions non-commutative symmetric functions

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Structure Constants for Immaculate functions

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ANNALS OF COMBINATORICS 2018年第2期22卷 347-361页

作者： Li, Shu Xiao York Univ Dept Math & Stat Toronto ON M3J 1P3 Canada

The immaculate functions, , were introduced as a Schur-like basis for NSym, the ring of noncommutative symmetric functions. We investigate their structure constants. These are analogues of Littlewood-Richardson coefficents. We will give a new proof of the left Pieri rule for the , a translation invariance property for the structure coefficients of the , and a counterexample to an -analogue of the saturation conjecture.

关键词： non-commutative symmetric functions Schur functions tableaux

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A PROBABILISTIC METHOD FOR THE NUMBER OF STANDARD IMMACULATE TABLEAUX

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ROCKY MOUNTAIN JOURNAL OF MATHEMATICS 2018年第6期48卷 2087-2097页

作者： Sun, Brian Y. Hu, Yingying Northeast Normal Univ Sch Math & Stat Changchun 130024 Jilin Peoples R China Northeast Normal Univ KLAS Changchun 130024 Jilin Peoples R China Xinjiang Univ Dept Math & Syst Sci Urumqi 830046 Peoples R China

In this paper, along the spirit of Greene, Nijenhuis and Wilf's probabilistic method for the classical hook-length formula for standard Young tableaux, we present a probabilistic proof of the hook-length formula f... 详细信息

关键词： Hook-length formula immaculate tableau probabilistic methods compositions non-commutative symmetric functions

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The Pieri Rule for Dual Immaculate Quasi-symmetric functions

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ANNALS OF COMBINATORICS 2016年第2期20卷 283-300页

作者： Bergeron, Nantel Sanchez-Ortega, Juana Zabrocki, Mike Fields Inst Res Math Sci 222 Coll StSecond Floor Toronto ON M5T 3J1 Canada York Univ Dept Math & Stat N York ON M3J 1P3 Canada Univ Toronto Dept Math 40 St George St Toronto ON M5S 2E4 Canada Univ Malaga Dept Algebra Geometry & Topol E-29071 Malaga Spain

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third authors to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here.

关键词： non-commutative symmetric functions quasi-symmetric functions tableaux Schur functions

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A non-commutative generalization of k-Schur functions

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DISCRETE MATHEMATICS 2009年第16期309卷 5092-5105页

作者： Bergeron, Nantel Descouens, Francois Zabrocki, Mike York Univ Dept Math & Stat Toronto ON M3J 1P3 Canada Fields Inst Toronto ON M5T 3J1 Canada

We introduce non-commutative analogs of k-Schur functions of Lapointe-Lascoux and Morse. We give explicit formulas for the expansions of non-commutative functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case. (C) 2009 Elsevier B.V. All rights reserved.

关键词： non-commutative symmetric functions k-Schur functions symmetric functions with additional parameters

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q and q, t-analogs of non-commutative symmetric functions

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DISCRETE MATHEMATICS 2005年第1-3期298卷 79-103页

作者： Bergeron, N Zabrocki, M York Univ Toronto ON M3J 1P3 Canada

关键词： non-commutative symmetric functions

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q and q, t-analogs of non-commutative symmetric functions

q and q, t-analogs of non-commutative s...

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14th International Conference on Formal Power Series and Algebraic Combinatorics

作者： Bergeron, N Zabrocki, M York Univ Toronto ON M3J 1P3 Canada

关键词： non-commutative symmetric functions

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Advanced determinant calculus: A complement

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LINEAR ALGEBRA AND ITS APPLICATIONS 2005年第1-3期411卷 68-166页

作者： Krattenthaler, C Univ Lyon 1 Inst Camille Jordan F-69622 Villeurbanne France

This is a complement to my previous article "Advanced Determinant Calculus" [C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (1999) ("The Andrews Festschrift"), Article B42q, 67 pp.]. In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory [G. Almkvist, C. Krattenthaler, J. Petersson, Some new formulas for pi, Experiment. Math. 12 (2003) 441-456]. Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. (c) 2005 Elsevier Inc. All rights reserved.

关键词： determinants vandermonde determinant Cauchy's double alternant skew circulant matrix confluent afternant confluent cauchy determinant pfaffian hankel determinants orthogonal polynomials chebyshev polynomials meixner polynomials laguerre polynomials continued fractions binomial coefficient catalan numbers fibonacci numbers bernoulli numbers stirling numbers non-intersecting lattice paths plane partitions tableaux rhombus tilings lozenge tilings alternating sign matrices non-crossing partitions perfect matchings permutations signed permutations inversion number major index compositions integer partitions descent algebra non-commutative symmetric functions elliptic functions the number pi LLL-algorithm

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