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The interval posets of permutations seen from the decomposition tree perspective

Mathilde Bouvel, Lapo Cioni, Benjamin Izart

Abstract

The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the decomposition trees of permutations, describing a procedure to obtain the former from the latter. We then give alternative proofs of some of the results in [B. Tenner, arXiv:2007.06142], and we solve the open problems that it posed (and some other enumerative problems) using techniques from symbolic and analytic combinatorics. Finally, we compute the Möbius function on such posets.

The interval posets of permutations seen from the decomposition tree perspective

Abstract

The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the decomposition trees of permutations, describing a procedure to obtain the former from the latter. We then give alternative proofs of some of the results in [B. Tenner, arXiv:2007.06142], and we solve the open problems that it posed (and some other enumerative problems) using techniques from symbolic and analytic combinatorics. Finally, we compute the Möbius function on such posets.

Paper Structure

This paper contains 12 sections, 18 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background: Interval posets and decomposition trees
  3. The interval poset of a permutation
  4. Substitution decomposition and decomposition trees
  5. Computing the poset from the decomposition tree
  6. Alternative proofs of known structural results
  7. Enumerative properties of interval posets
  8. The number of generators of a given interval poset
  9. Interval posets with exactly two generators
  10. Counting interval posets
  11. Counting tree interval posets
  12. The Möbius function on interval posets

Key Result

Proposition 4

For any permutation $\sigma$, it holds that $\mathop{\mathrm{inv}}\nolimits(\bar{P}(\sigma)) = \tilde{P}(\sigma^{-1})$ as plane embeddings. Consequently, we also have $\mathop{\mathrm{inv}}\nolimits(\bar{P}_{\bullet}(\sigma))=\tilde{P}_{\bullet}(\sigma^{-1})$ (with obvious extension of the notation

Figures (4)

  • Figure 1: From left to right: The canonical embedded poset $\bar{P}(\sigma)$, and the standard embedded poset $\tilde{P}(\sigma)$, for $\sigma = 4\, 5\, 6\, 7\, 9\, 3\, 1\, 2\, 8$.
  • Figure 2: From left to right: The standard embedded poset $\tilde{P}(\sigma^{-1})$, and the decomposition tree $T(\sigma^{-1})$, for $\sigma = 4\, 5\, 6\, 7\, 9\, 3\, 1\, 2\, 8$, i.e., $\sigma^{-1} = 7\, 8\, 6\, 1\, 2\, 3\, 4\, 9\, 5$. The substitution decomposition of $\sigma^{-1}$ is indeed $\sigma^{-1}= 3142[\ominus[\oplus[1,1],1],\oplus[1,1,1,1],1,1]$.
  • Figure 3: From left to right: an embedding (canonical or standard, since they are the same) of the dual claw poset of size $5$ and of the argyle posets of size $3$ and $5$ (the labeling of their nodes has been omitted).
  • Figure 4: The Möbius function between the maximum and any element in an argyle poset.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Theorem 5
  • Remark 6
  • Definition 7
  • Theorem 8
  • Proposition 9
  • ...and 34 more