Parts and Subword Patterns in Compositions
Journal of Combinatorics and Number Theory
We find relationships between subword patterns and residue classes of parts in the set of integer compositions of a given weight. In particular, we show that it is always possible to express the total number of parts in compositions of n that are congruent to i modulo m as a linear combination of the total number of occurrences of subword patterns of length no more than m. We also find an explicit formula enumerating all such parts.
Hopkins, Brian, Mark Shattuck, Andrew V. Sills, Thotsaporn Thanatipanonda, Hua Wang.
"Parts and Subword Patterns in Compositions."
Journal of Combinatorics and Number Theory, 9 (1): 1-14.