Lukas Riegler - Generalized monotone triangles

dmtcs:2331 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2331
Generalized monotone trianglesArticle

Authors: Lukas Riegler 1

  • 1 Fakultät für Mathematik [Wien]

In a recent work, the combinatorial interpretation of the polynomial $\alpha (n; k_1,k_2,\ldots,k_n)$ counting the number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ was extended to weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n$. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles – a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of $\alpha (n; k_1,k_2,\ldots,k_n)$ at arbitrary $(k_1,k_2,\ldots,k_n) ∈ \mathbb{Z}^n$ is a signed enumeration of Generalized Monotone Triangles with bottom row $(k_1,k_2,\ldots,k_n)$. Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving bijective proofs.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Combinatorial Reciprocity,Monotone Triangle,Generalized Monotone Triangle,Alternating Sign Matrix,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Compact enumeration formulas for generalized partitions; Code: Y 463

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