Joint equidistributions of mesh patterns 123 and 321 with symmetric and minus-antipodal shadings

Abstract

In this paper, we extend recent results by Lv and Kitaev by proving 20 (out of 22 possible) joint equidistributions of mesh patterns 123 and 321 with symmetric shadings, as well as all 36 joint equidistributions of these patterns with minus-antipodal shadings. Our results link several joint equidistributions of mesh patterns to various integer sequences, including unsigned Stirling numbers of the first kind, harmonic numbers, and the numbers of inversion sequences avoiding a certain vincular pattern studied by Lin and Yan.

Figures (4)

• Figure 1: An example of the permutation $14325$ avoiding $p$, where the circled dots in the diagrams represent the respective occurrences of the pattern $123$.
• Figure 2: The structure of Nr. 17
• Figure 3: The structure of $n$-permutations containing occurrences of $q_1$ and $q_2$ in Nr. S19, where the $x_i$'s represent the left-to-right minima in permutations.
• Figure 4: The swapping operation that maps $\pi^{(i)}$ with the lexicographically first occurrence of $q_2$ (on the left side) to $\pi^{(i+1)}$ with one occurrence of $q_1$ (on the right side) in Theorem \ref{['thm-pairs-21-22']}.

Theorems & Definitions (20)

• Theorem 1
• proof
• Conjecture 2
• Theorem 3
• proof
• Theorem 4
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm-pairs-39-47']}
• Remark