Alternating Subsets and Successions

Abstract

We present a unified extension of alternating subsets to \(k\)-combinations
of \(\{1, 2, \ldots, n\}\) containing a prescribed number of sequences
of elements of the same parity. This is achieved by shifting attention
from parity-alternating elements to pairs of adjacent elements of the
same parity.

Enumeration formulas for both linear and circular combinations are
obtained by direct combinatorial arguments. The results are applied
to the enumeration of bit strings.