On Partitions with Palindromic Gap Sequences

Abstract

We introduce and study palindromic-gap partitions: partitions whose sequence of successive differences between consecutive parts is a palindrome. We derive the generating function for this family, treating partitions with an odd and an even number of parts separately, and obtain a bivariate refinement tracking the number of parts. Specializing further, we give explicit closed formulas for the number of palindromic-gap partitions of $n$ into exactly $r$ parts. We then construct explicit bijections between palindromic-gap partitions with $2m+1$, respectively $2m$, parts and ordinary partitions with at most $m+1$ parts, in which the weight of a palindromic-gap partition is encoded by the largest part, or the two largest parts, of its image; composing these yields an explicit bijection between the odd and even families themselves. Finally, we show that a partition is palindromic-gap if and only if its Ferrers diagram, drawn with left-justified rows, is self-complementary under a $180^\circ$ rotation inside its own naturally associated rectangle, identifying palindromic-gap partitions with self-complementary partitions relative to this rectangle. This yields a partial answer to a question of Keith on the enumeration of complementable and self-complementary partitions: we give explicit generating functions for the self-complementary case and exhibit, at $n=15$, a partition that is complementable but not self-complementary, confirming that the two notions are genuinely distinct.