Boolean--Eulerian numbers

TL;DR

This paper explores decreasing binary trees with colored vertices, establishing bijections to set partitions and 1-2 trees, and introduces Boolean--Eulerian polynomials with properties like gamma-positivity and real-rootedness.

Contribution

It constructs new bijections linking decreasing binary trees to set partitions and 1-2 trees, and introduces Boolean--Eulerian polynomials with algebraic and combinatorial properties.

Findings

Exponential generating function is 1/(1−tan z).

Count of trees equals 2^{n−1} times the Euler number.

Boolean--Eulerian polynomials are algebraic transforms of Eulerian polynomials.

Abstract

We study decreasing binary trees in which every vertex with two children is colored red or blue. We construct two bijections. The first, to ordered set partitions into odd-sized blocks each arranged as an alternating permutation, shows that the exponential generating function of these trees is $1/(1-\tan z)$. The second, to nonplane decreasing 1-2 trees paired with a binary label on each non-root vertex, proves combinatorially that the count equals $2^{n-1}$ times the~$n$th Euler number. Refining by the number of right edges yields the Boolean--Eulerian polynomials, which are an explicit algebraic transform of the classical Eulerian polynomials. The Foata--Strehl orbit decomposition, recast in the decreasing-binary-tree model, gives a direct combinatorial proof of gamma-positivity, and the algebraic transform carries real-rootedness and interlacing of zeros from the Eulerian polynomials to the Boolean--Eulerian polynomials.