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Real-rootedness of rook-Eulerian polynomials

Per Alexandersson, Aryaman Jal, Maena Quemener

TL;DR

The note introduces rook-Eulerian polynomials $Q^{\\lambda}(t)$ arising from complete rook placements on Ferrers boards and proves their real-rootedness using interlacing techniques, while connecting these polynomials to lower Bruhat intervals of $312$-avoiding permutations. It extends to a multivariate refinement $Q^{\\lambda}(\\mathbf{x})$ that is same-phase stable and to a row-complete rook-placement setting with a corresponding recursion, illustrating the method's versatility. The work also situates rook-Eulerian polynomials relative to $\\mathbf{s}$-Eulerian polynomials, showing they form a distinct real-rooted family, and explores broader contexts including descents in Bruhat intervals and multiset rook-Eulerian polynomials, yielding several non-real-rooted counterexamples and conjectures. Together, these results establish a new, structurally rich Eulerian-type family tied to rook theory and permutation posets, with multiple natural generalizations and open questions about stability and real-rootedness. The findings thus broaden the toolkit for real-rooted combinatorial polynomials and motivate further study of stability phenomena in related poset and rook-theoretic settings.

Abstract

We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to interpret these rook placements is as lower intervals of $312$-avoiding permutations in the Bruhat order. We end with some variations and generalizations along this theme.

Real-rootedness of rook-Eulerian polynomials

TL;DR

The note introduces rook-Eulerian polynomials arising from complete rook placements on Ferrers boards and proves their real-rootedness using interlacing techniques, while connecting these polynomials to lower Bruhat intervals of -avoiding permutations. It extends to a multivariate refinement that is same-phase stable and to a row-complete rook-placement setting with a corresponding recursion, illustrating the method's versatility. The work also situates rook-Eulerian polynomials relative to -Eulerian polynomials, showing they form a distinct real-rooted family, and explores broader contexts including descents in Bruhat intervals and multiset rook-Eulerian polynomials, yielding several non-real-rooted counterexamples and conjectures. Together, these results establish a new, structurally rich Eulerian-type family tied to rook theory and permutation posets, with multiple natural generalizations and open questions about stability and real-rootedness. The findings thus broaden the toolkit for real-rooted combinatorial polynomials and motivate further study of stability phenomena in related poset and rook-theoretic settings.

Abstract

We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to interpret these rook placements is as lower intervals of -avoiding permutations in the Bruhat order. We end with some variations and generalizations along this theme.

Paper Structure

This paper contains 8 sections, 9 theorems, 37 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Real-rootedness of rook-Eulerian polynomials
  3. A multivariate generalization
  4. Variations and generalizations
  5. Relation to s-Eulerian polynomials
  6. Descents in general Bruhat intervals
  7. Multiset rook-Eulerian polynomials
  8. Acknowledgements

Key Result

Proposition 3

Every $312$-avoiding permutation $\sigma \in \mathfrak{S}_n$ corresponds to a Ferrers board $\lambda = (\lambda_1,\dotsc,\lambda_n)$ with $\lambda_i \geq i$ for $i=1, \ldots, n-1$ and $\lambda_{n} = n$, and vice-versa. The permutations in the interval are exactly the complete rook placements on the Ferrers board $\lambda$.

Figures (3)

  • Figure 1: The complete rook placement $\sigma = 2431$ on $\lambda = 3444$.
  • Figure 2: Left: The $312$-pattern in a rook configuration. Right: The complete rook placement on the Ferrers board $45566888$ corresponds to the 312-avoiding permutation $45362871$. Every rook placement on this board can be obtained by switching pairs of non-nested rooks in the placement above.
  • Figure 3: The bijection between rook placements on a Ferrers board $\lambda = 2344$ and an interval $\{\sigma \in \mathfrak{S}_{4}: \text{id} \leq_{B} \sigma_{B} \leq 2341 \}$ in the Bruhat order. Black edges represent cover relations corresponding to arbitrary transpositions --- or applications of the switch move at the level of the rooks --- while yellow edges represent cover relations corresponding to adjacent transpositions.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2: Weak and strong order
  • Proposition 3
  • proof
  • Example 4
  • Definition 5
  • Example 6
  • Example 7
  • Definition 8
  • Definition 9
  • ...and 25 more