부산대학교 도서관

In 1980, D.~M. Bressoud conjectured a very general partition identity that includes many classical results [Mem. Amer. Math. Soc. {\bf 24} (1980), no.~227, 54 pp; MR0556608]. Bressoud's conjecture depends on non-negative integer parameters $\lambda$, $\eta$, $k$, and $r$, as well as a choice of $\lambda$ residue classes modulo $\eta$, but for the purposes of this review we suppress all of this and simply state the conjecture as $B_j(n) = A_j(n)$, for $j \in \{0,1\}$, where $B_j(n)$ counts the number of partitions of $n$ satisfying (among other things) certain difference conditions and $A_j(n)$ counts the number of partitions of $n$ whose parts lie in certain congruence classes. Bressoud's conjecture was known in some special cases and is now known in full for $j=1$ thanks to recent work of S. Kim [Adv. Math. {\bf 325} (2018), 770--813; MR3742602]. \par In this paper, the authors announce a forthcoming proof of the case $j=0$ and present the first major step in the argument. Perhaps unexpectedly, a key role is played by overpartitions. Specifically, the authors define overpartition analogues of the counting functions $B_j(n)$ and $A_j(n)$, denoted $\overline{B}_j(n)$ and $\overline{A}_j(n)$, and then prove that the generating function for $\overline{B}_j(n)$ is an infinite product multiple of the generating function for $B_{1-j}(n)$. The proof takes up over forty pages and involves advanced and novel combinatorial arguments. Using known cases of $B_j(n) = A_j(n)$ then leads to families of overpartition identities, both known and new. \par In the last part of the paper, the authors use Bailey pairs to prove a $q$-hypergeometric series identity which is expected to be the generating function version of the identity $\overline{B}_0(n) = \overline{A}_0(n)$. They close by previewing Part II of the series, wherein they prove that $\overline{B}_1(n) = \overline{A}_1(n)$. Combined with the main result of Part I, this settles Bressoud's conjecture. \par Regarding the role of overpartitions in their work, the authors write, ``It should be stressed that the overpartition analogues considered in this paper are not merely a matter of extension and specialization. In fact, they play an essential role and serve as an indispensable structure in tackling the conjecture of Bressoud formulated in terms of ordinary partitions.''