Enumeration by kernel positions for strongly Bernoulli type truncation games on words

TL;DR

This paper develops a combinatorial approach to truncation games on words, deriving new formulas for Bernoulli numbers and polynomials, and providing a novel model for counting connected permutations.

Contribution

It introduces a new enumeration method based on kernel positions for a class of truncation games, linking combinatorial game theory with number theory and permutation enumeration.

Findings

Derived new formulas for Bernoulli numbers and polynomials of the second kind.

Established a bijective correspondence between the game decomposition and King's decomposition.

Provided a combinatorial model for counting connected permutations of a given rank.

Abstract

We find the winning strategy for a class of truncation games played on words. As a consequence of the present author's recent results on some of these games we obtain new formulas for Bernoulli numbers and polynomials of the second kind and a new combinatorial model for the number of connected permutations of given rank. For connected permutations, the decomposition used to find the winning strategy is shown to be bijectively equivalent to King's decomposition, used to recursively generate a transposition Gray code of the connected permutations.