Parity results for higher order SPT-functions of overpartitions

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发表日期：

2024年12月26日

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Weifeng Gao,
Eric H. Liu,
Olivia X. M. Yao

摘要

In 2015, Jennings-Shaffer defined the higher order SPT-functions for overpartitions $\overline{spt}_k(n) $ ${\bar{spt}}_{k} (n)$ and gave the combinatorial interpretation of $\overline{spt}_k(n) $ ${\bar{spt}}_{k} (n)$ . In recent years, some congruences for $\overline{spt}_k(n)$ ${\bar{spt}}_{k} (n)$ have been proved. Garvan and Jennings-Shaffer presented a characterization of the parity on $\overline{spt}_1( n)$ ${\bar{spt}}_{1} (n)$ . Motivated by their work, in this paper, we give a characterization of congruences modulo 2 on $\overline{spt}_2( n)$ ${\bar{spt}}_{2} (n)$ and prove a congruence modulo 4 for $\overline{spt}_2( n)$ ${\bar{spt}}_{2} (n)$ and several parity results for $\overline{spt}_3( n)$ ${\bar{spt}}_{3} (n)$ by using the generating functions of $\overline{M}(r,8,n) $ $\bar{M} (r, 8, n)$ which denote the number of overpartitions of n whose first residual crank is congruent to r modulo 8.