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Substring compatibility of permutation statistics

Michael Tang

TL;DR

The paper studies permutation statistics that are both shuffle-compatible and substring-compatible, introducing a substring coalgebra $\mathcal{C}_{\textnormal{st}}$ and, under a weak shuffle-compatibility assumption, forming a Hopf algebra $\mathcal{H}_{\textnormal{st}}$ that encompasses known examples. It shows that the descent set and peak set give Hopf-algebra realizations isomorphic to $\textnormal{QSym}$ and the algebra of peaks $\Pi$, respectively, and proves that these statistics are bicompatible. A central conjecture posits that, up to equivalence, the only nontrivial bicompatible statistics are the descent, peak, valley, and trivial statistics, with partial progress establishing that any other bicompatible statistic must be trivial on permutations of length at most $6$. The paper also derives a number-theoretic restriction that constrains nontrivial extensions and uses SAT-solver computations to show nonexistence of additional bicompatible statistics up to length $5$, implying triviality at length $6$ when lengths are prime powers are considered. Overall, this work unifies shuffle- and substring-based invariants into a Hopf-algebra framework and advances the classification of permutation statistics under joint compatibility constraints, with potential implications for understanding connections to $\textnormal{SSym}$, $\textnormal{QSym}$, and $\Pi$.

Abstract

A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture.

Substring compatibility of permutation statistics

TL;DR

The paper studies permutation statistics that are both shuffle-compatible and substring-compatible, introducing a substring coalgebra and, under a weak shuffle-compatibility assumption, forming a Hopf algebra that encompasses known examples. It shows that the descent set and peak set give Hopf-algebra realizations isomorphic to and the algebra of peaks , respectively, and proves that these statistics are bicompatible. A central conjecture posits that, up to equivalence, the only nontrivial bicompatible statistics are the descent, peak, valley, and trivial statistics, with partial progress establishing that any other bicompatible statistic must be trivial on permutations of length at most . The paper also derives a number-theoretic restriction that constrains nontrivial extensions and uses SAT-solver computations to show nonexistence of additional bicompatible statistics up to length , implying triviality at length when lengths are prime powers are considered. Overall, this work unifies shuffle- and substring-based invariants into a Hopf-algebra framework and advances the classification of permutation statistics under joint compatibility constraints, with potential implications for understanding connections to , , and .

Abstract

A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture.

Paper Structure

This paper contains 19 sections, 20 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Shuffle compatibility
  3. Permutations and permutation statistics
  4. Refinement of permutation statistics
  5. Shuffle compatibility and the shuffle algebra
  6. Weak shuffle compatibility
  7. Substring compatibility
  8. Lifts of permutation statistics
  9. The substring coalgebra
  10. Bicompatibility
  11. The descent set and peak set
  12. Characterizing bicompatible permutation statistics
  13. A note on weakly bicompatible statistics
  14. The descent set
  15. The peak set and valley set
  16. ...and 4 more sections

Key Result

Proposition 2.11

A permutation statistic $\textnormal{st}$ is shuffle-compatible if and only if whenever $\pi$, $\pi'$, $\sigma$, and $\sigma'$ are permutations with $\pi \sim_\textnormal{st} \pi'$ and $\sigma \sim_\textnormal{st} \sigma'$, we have $\pi \shuffle \sigma \sim_\textnormal{st} \pi' \shuffle \sigma'$.

Figures (1)

  • Figure 1: The maps between $\textnormal{SSym}$, $\textnormal{QSym}$, $\Pi$, and the corresponding Hopf algebras of permutation statistics $\mathcal{H}_{\mathop{\mathrm{dis}}\nolimits}$, $\mathcal{H}_{\mathop{\mathrm{Des}}\nolimits}$, and $\mathcal{H}_{\mathop{\mathrm{Pk}}\nolimits}$.

Theorems & Definitions (64)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5: gessel-zhuang
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • Proposition 2.11
  • ...and 54 more