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Probabilistic Entry Swapping Bijections for Non-Attacking Fillings

Guilherme Zeus Dantas e Moura, Olya Mandelshtam

TL;DR

This work addresses symmetry properties of permuted-basement Macdonald polynomials by developing a fully combinatorial, probabilistic bijection on non-attacking fillings. It generalizes Mandelshtam's partition-focused bijection to composition shapes, enabling a weight-preserving swap of adjacent basement entries that underpins the symmetry E_α^σ(x;q,t) = E_α^{σ s_i}(x;q,t). The authors prove this symmetry for α_i = α_{i+1} without requiring σ_i and σ_{i+1} to be consecutive, thereby extending prior results from Ale19 and CHMMW22. The approach relaxes earlier hypotheses and provides a unified combinatorial framework for composition-shaped non-attacking fillings with potential broader implications in Macdonald polynomial theory.

Abstract

Non-attacking fillings are combinatorial objects central to the theory of Macdonald polynomials. A probabilistic bijection for partition-shaped non-attacking fillings was introduced by Mandelshtam (2024) to prove a compact formula for symmetric Macdonald polynomials. In this work, we generalize this probabilistic bijection to composition-shaped non-attacking fillings. As an application, we provide a bijective proof to extend a symmetry theorem for permuted-basement Macdonald polynomials established by Alexandersson (2019), proving a version with fewer assumptions.

Probabilistic Entry Swapping Bijections for Non-Attacking Fillings

TL;DR

This work addresses symmetry properties of permuted-basement Macdonald polynomials by developing a fully combinatorial, probabilistic bijection on non-attacking fillings. It generalizes Mandelshtam's partition-focused bijection to composition shapes, enabling a weight-preserving swap of adjacent basement entries that underpins the symmetry E_α^σ(x;q,t) = E_α^{σ s_i}(x;q,t). The authors prove this symmetry for α_i = α_{i+1} without requiring σ_i and σ_{i+1} to be consecutive, thereby extending prior results from Ale19 and CHMMW22. The approach relaxes earlier hypotheses and provides a unified combinatorial framework for composition-shaped non-attacking fillings with potential broader implications in Macdonald polynomial theory.

Abstract

Non-attacking fillings are combinatorial objects central to the theory of Macdonald polynomials. A probabilistic bijection for partition-shaped non-attacking fillings was introduced by Mandelshtam (2024) to prove a compact formula for symmetric Macdonald polynomials. In this work, we generalize this probabilistic bijection to composition-shaped non-attacking fillings. As an application, we provide a bijective proof to extend a symmetry theorem for permuted-basement Macdonald polynomials established by Alexandersson (2019), proving a version with fewer assumptions.

Paper Structure

This paper contains 6 sections, 4 theorems, 24 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutations and compositions
  4. Permuted-basement Macdonald polynomials
  5. Symmetry theorem for permuted-basement Macdonald polynomials
  6. Non-attacking fillings

Key Result

Theorem 1.1

Let $\alpha= \readlist\thecycle{\alpha_1,\ldots,\alpha_n} \foreachitem\i\in\thecycle{$ be a composition with $\alpha_i=\alpha_{i+1}$ for some $i\in[n-1]$. Let $\sigma \in \mathscr{S}_{n}$, such that $\sigma_{i+1} = \sigma_i \pm 1$. Then,

Figures (5)

  • Figure 1: The left action of $\pi = \left[ \readlist\thecycle{3, 1, 2, 4, 6, 5} \foreachitem\i\in\thecycle{\i} \right]$ on $\alpha = \readlist\thecycle{3, 1, 0, 2, 2, 0} \foreachitem\i\in\thecycle{$.
  • Figure 2: The skyline diagram (left) and the augmented skyline diagram (right) of $\alpha = \readlist\thecycle{2, 3, 0, 1, 2, 0} \foreachitem\i\in\thecycle{$. The basement of the augmented skyline diagram is shaded.
  • Figure 3: We show set of boxes counted by $\mathop{\mathrm{leg}}\nolimits(u)$ ( ), the left-arm set ( ), the right-arm set ( ), and the south ( ) of a box $u = (8, 2)$ ( ) in the augmented skyline diagram of $\alpha = \readlist\thecycle{3, 2, 2, 4, 4, 0, 3, 3, 3, 4, 2, 1, 3} \foreachitem\i\in\thecycle{$. Then $\mathop{\mathrm{leg}}\nolimits(u)=1$ and $\mathop{\mathrm{arm}}\nolimits(u)=5$.
  • Figure 4: Configuration of attacking boxes in a same row (left) and in consecutive rows (right).
  • Figure 5: A non-attacking filling of shape $\alpha = \readlist\thecycle{2, 2, 0, 1} \foreachitem\i\in\thecycle{$, basement $\sigma = \left[ \readlist\thecycle{3, 1, 2, 4} \foreachitem\i\in\thecycle{\i} \right]$, and $\mathop{\mathrm{content}}\nolimits(T) = \readlist\thecycle{1, 2, 0, 2} \foreachitem\i\in\thecycle{$.

Theorems & Definitions (15)

  • Theorem 1.1: equivalent to Ale19
  • Theorem 1.2
  • Example 2.1
  • Remark 2.2
  • Definition 2.3: Fer11
  • Remark 2.4
  • Example 2.5
  • Definition 2.6: Permuted-basement Macdonald polynomials
  • Remark 2.7
  • Remark 2.8
  • ...and 5 more