Intervals in Dyck paths and the wreath conjecture

TL;DR

The paper counts intervals of length $m$ containing exactly $l$ falls across all Dyck paths of semilength $k$, denoted $\iota_k(m,l)$. It derives a closed-form expression for $\iota_k(m,l)$ in terms of binomial coefficients, including an alternative form, and proves the special case $\iota_k(k,l)=\binom{k}{l}^2$. Motivated by this combinatorial data, the authors propose strengthened variants of Baranyai–Katona's wreath conjecture for $n=2k+1$ and show that the corollary provides a necessary condition for these conjectures; they verify the conjectures computationally for small $k$ and develop a NE upper-path and ballot-number framework to support the enumeration results.

Abstract

Let $ι_{k}(m,l)$ denote the total number of intervals of length $m$ across all Dyck paths of semilength $k$ such that each interval contains precisely $l$ falls. We give the formula for $ι_{k}(m,l)$ and show that $ι_{k}(k,l)=\binom{k}{l}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.

Figures (5)

• Figure 1: Examples of permutations which confirm Conjectures \ref{['conj:weaker']} and \ref{['conj:stronger']} for $k \leq 3$. Below each Dyck $k$-path the corresponding permutation $\pi$ is written in the form $\pi(0), \pi(1), \dots, \pi(2k)$. The second conditions of the conjectures require numbers $1,2,\dots, k$ to lie below rises and numbers $k+1, k+2, \dots, 2k$ below falls.
• Figure 2: The bijection from \ref{['lem:main']} given by 'flipping' the section of the path after the highlighted point.
• Figure 3: An illustration of NE upper paths whose interval of length $m$ starting at $(i,i+d)$ contains exactly $l$ East steps.
• Figure 4: An illustration of the left-hand side of equation \ref{['eq:claim']}.
• Figure 5: The bijection from \ref{['Claim']}. The additional step in $W'$ is the East step leading to $(-i,i)$.

Theorems & Definitions (13)

• Theorem 1.1
• Corollary 1.2
• Conjecture 2.1: The wreath conjecture
• Conjecture 2.2
• Lemma 2.3
• proof
• Conjecture 2.4
• Lemma 3.1
• proof
• proof : Proof of \ref{['thm:main']}