The parity of the number of missing integers in partitions

Abstract

Missing integers in partitions have been studied by many authors. A missing integer of a partition is a positive integer smaller than its largest part that does not occur as a part. We investigate parity biases for the number of missing integers in partitions and overpartitions. For every integer $c\ge2$, we prove that the coefficients $a_c(n)$ of $(-cq;q)_\infty/(-q;q)_\infty$ are eventually strictly positive and provide an effectively computable threshold. The proof uses the circle method, combining a saddle-point analysis on the major arc with explicit uniform estimates on the minor arcs. Specializing to $c=2$ and $c=3$, and verifying the remaining finite cases, we show that partitions and overpartitions of $n$ with an even number of missing integers outnumber those with an odd number for $n>34$ and $n>27$, respectively. We further prove that, for $n>10$, the difference between the numbers of ordinary partitions with an even and an odd number of missing integers is smaller than the number of partitions of $n$ into distinct parts. These results confirm three conjectures of Bhoria, Eyyunni and Santra. We also give a combinatorial interpretation of $a_c(n)$ for every integer $c\ge4$ in terms of colored partitions, generalizing the partition and overpartition interpretations of Bhoria, Eyyunni and Santra for $c=2$ and $c=3$, respectively.