An assortment of problems in permutation patterns: unimodality, equivalence, derangements, and sorting

Abstract

We collect open problems in permutation patterns on four themes: rank-unimodality in the permutation pattern poset, Wilf-equivalence and shape-Wilf-equivalence, the enumeration of derangements in permutation classes, and sorting by stacks in series, generalized stacks, and restricted containers (C-machines).

Figures (4)

• Figure 1: The permutation $\sigma = 32514$ (left) is contained in $\pi = 362957184$ (right). The circled entries in $\pi$ form the subsequence $32918$, which is order-isomorphic to $\sigma$. The permutation $\pi$ avoids $4321$ because it has no decreasing subsequence of length four.
• Figure 2: The symmetries of the square, labelled by their effect on a permutation $\pi$.
• Figure 3: The proof of Proposition \ref{['prop:swe-sums']} with $\alpha = 231$, $\beta = 312$, and $\gamma = 21$. The shaded region is the shadow cast by copies of $21$. The bijection labeled BS is due to Bloom and Saracino bloom:a-simple-biject:.
• Figure 4: The proportion of $\beta$-avoiding permutations that are derangements, by length.

Theorems & Definitions (18)

• Conjecture 2.1
• Conjecture 2.2: McNamara and Steingrímsson mcnamara:on-the-topology:
• Proposition 2.3
• Theorem 2.4: Sagan sagan:compositions-in:
• proof
• Corollary 2.5
• Conjecture 2.6
• Proposition 3.2
• Theorem 3.3: Backelin, West, and Xin backelin:wilf-equivalenc:
• Theorem 3.4: Stankova and West stankova:a-new-class-of-: