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Deficit and $(q,t)$-symmetry in triangular partitions

Viviane Pons, Loïc Le Mogne

Abstract

We study the $(q,t)$-enumeration of triangular Dyck paths considered by Bergeron and Mazin. To do so, we introduce the notion of triangular and sim-sym tableaux and the deficit statistic which is a new interpretation of the dinv. We use it to obtain new results and proofs on triangular $2$-partitions and an interesting conjecture for a certain lattice interval $(q,t,r)$-enumeration.

Deficit and $(q,t)$-symmetry in triangular partitions

Abstract

We study the -enumeration of triangular Dyck paths considered by Bergeron and Mazin. To do so, we introduce the notion of triangular and sim-sym tableaux and the deficit statistic which is a new interpretation of the dinv. We use it to obtain new results and proofs on triangular -partitions and an interesting conjecture for a certain lattice interval -enumeration.

Paper Structure

This paper contains 23 sections, 21 theorems, 29 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Partitions, Young tableaux and Schur functions
  4. Triangular partitions
  5. Area, sim and (q,t)-enumeration
  6. Combinatorial interpretation of similar cells
  7. Triangular Tableau and deficit
  8. Deficit of a sub-partition
  9. Sim-Sym Tableau
  10. Sim-sym tableaux and $(q,t)$ enumeration on $2$-partitions
  11. Characterization of triangular $2$-partitions
  12. Row-regulars tableaux
  13. Explicit $(q,t)$-enumeration and Schur positivity
  14. Case $a=d$
  15. Case $d < a \leq m - d - i +2$
  16. ...and 8 more sections

Key Result

Lemma 2.6

Let $\lambda$ be a partition, then $\lambda$ is triangular if and only if $v_\lambda^{-} < v_\lambda^{+}$. Then for all $v \in ]v_\lambda^{-}, v_\lambda^{+}[$, $\lambda$ is cut off by a line of slope vector $v$. $\blacktriangleleft$$\blacktriangleleft$

Figures (19)

  • Figure 1: The Ferrers diagram of $(4,3,1)$ and an example of standard Young tableau.
  • Figure 2: Some Young tableaux for the partition $(3,3)$.
  • Figure 3: Semi-standard Young tableaux corresponding to some Schur polynomials.
  • Figure 4: Left: the triangular partition $(4,3,1)$ with one line that cuts it off and its slope vector. Right: a non-triangular partition $(4,4)$.
  • Figure 5: Left: triangular Dyck path $(7,6,4,3,1), (5,5,3,2)$ with ${\mathop{\mathrm{area}}\nolimits_{\lambda} (\mu) = 6}$ (cells in light red), $\mathop{\mathrm{sim}}\nolimits_{\lambda} (\mu) = 13$ (cells in light green), and $\mathop{\mathrm{def}}\nolimits_{\lambda} (\mu) = 2$ (dotted yellow cells). Right: the same triangular Dyck path where the hook of a given cell is shown.
  • ...and 14 more figures

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: From Berg1
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: Lemma 1.2 of Berg1
  • Definition 2.7
  • Definition 2.8: From Berg1, Section 4.1
  • Remark 2.9
  • Conjecture 1: Conjecture 1 of Berg1
  • ...and 47 more