Black-White Cell Capacity in $k$-ary Words and Permutations

We introduce a new bargraph statistic that we call black-white cell capacity. It is obtained by coloring the cells of the bargraph in a chessboard style and recording the numbers of black and white cells contained in the bargraph. We study two word families under this statistic: $k$-ary words and permutations. We obtain the corresponding generating function, in the $k$-ary words case, and a closed-form formula for each $n$, in the permutations case. Of special interest are words containing an equal number of black and white cells, that we call bw-balanced. We obtain generating functions, closed-form formulas, and asymptotics in both cases.