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

We find the winning strategy for a class of truncation games played on words. As a consequence of the present authorʹs 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 Kingʹs decomposition, used to recursively generate a transposition Gray code of the connected permutations.