On Partitions with Constant-Sum Gap Sequences

Abstract

We introduce and study \emph{constant-sum gap partitions}: for a fixed nonnegative integer $C$, these are partitions $\lambda =({\lambda }_1,\dots ,{\lambda }_k)$ whose successive differences $d_i={\lambda }_{i+1}-{\lambda }_i$ satisfy $d_i+d_{k-i}=C$ for all valid $i$. We derive the generating function for this family, treating partitions of odd and even length separately, and compute it explicitly for $C=0,1,2,3,4$. We then construct explicit bijections between constant-sum gap partitions with $2m+1$, respectively $2m$, parts and partitions into odd, respectively even, parts in which parts below the largest part have bounded multiplicity, together with an exact weight relation between a partition and its image. Finally, we give a geometric characterization of constant-sum gap partitions in terms of their Ferrers diagrams: removing a prescribed, linearly growing number of cells from each row renders the diagram symmetric about its middle row, for both the odd- and even-length cases.

arXiv comment24 pages