Boolean-Narayana numbers

TL;DR

The paper defines Boolean-Narayana numbers $BoNa(n,k)$ as the refinement of Boolean-Catalan numbers counting 0-1-trees on $n$ vertices with $k-1$ right edges, and provides two combinatorial interpretations via stack-sorting permutations and pattern-avoiding classes. It delivers an explicit formula $BoNa(n,k)=\frac{1}{n}\binom{n}{k-1}\sum_{j=0}^{\min(k-1,n-k)} 2^j \binom{k-1}{j}\binom{n-k+1}{j+1}$ using Lagrange Inversion, along with proofs of unimodality in $k$ for fixed $n$, and that the generating polynomials $BoNa_n(u)$ have only real zeros, yielding horizontal log-concavity; it also proves vertical log-concavity in $n$ for fixed $k$ via a binomial-transform argument. The work links Boolean-Narayana numbers to Narayana polynomials and permutation-pattern classes, enriching Narayana-type refinements for tree-based combinatorial structures and pointing to broader bijective connections with permutation classes and other tree families.

Abstract

We introduce a refinement of Boolean-Catalan numbers and call them Boolean-Narayana numbers. We provide an explicit formula for these numbers, and prove unimodality, log-concavity, and real-roots-only results for their sequences.

Figures (3)

• Figure 1: The six 0-1 trees on three vertices.
• Figure 2: A 0-1 tree $T$ and its image $f(T)$.
• Figure 3: The action of $z$ on a tree in $\mathcal{B}(9,4)$.

Theorems & Definitions (13)

• Definition 1.1
• Proposition 2.1
• Proposition 2.2
• Corollary 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1
• Example 4.2
• Lemma 4.3
• Theorem 5.1