Alternating Sums in Compositions
Runde Gao, Jia Huang
Abstract
Motivated by the recent asymptotic formula for the moment of the alternating sum of parts in partitions by Craig and Li, we study the moment of the alternating sum of parts in compositions. We establish closed formulas for this moment using techniques involving generating functions, matrix diagonalization, Krawtchouk polynomials, differential equations, and Faà di Bruno's formula. Our result is connected with some known integer sequences in the OEIS and admits a combinatorial interpretation by coin flipping.
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