N. Bergeron ; F. Descouens ; M. Zabrocki - A generalization of $(q,t)$-Catalan and nabla operators

dmtcs:3597 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) - https://doi.org/10.46298/dmtcs.3597
A generalization of $(q,t)$-Catalan and nabla operatorsArticle

Authors: N. Bergeron 1; F. Descouens 2,1; M. Zabrocki 2

  • 1 Department of Mathematics and Statistics [Toronto]
  • 2 Fields Institute for Research In Mathematical Sciences

We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths.

https://doi.org/10.46298/dmtcs.3597
Source: HAL:hal-01185130v1
Volume: DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
Section: Proceedings
Published on: January 1, 2008
Imported on: May 10, 2017
Keywords: $k$-Schur functions,nabla operator,$(q t)$-Catalan numbers,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Funder: Natural Sciences and Engineering Research Council of Canada

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