Two Families of Buffered Frobenius Representations of Overpartitions

Abstract

We generalize the generating series of the Dyson ranks and M 2ranks of overpartitions to obtain k-fold variants, and give a combinatorial interpretation of each. The k-fold generating series correspond to the full ranks of two families of buffered Frobenius representations, which generalize Lovejoy's first and second Frobenius representations of overpartitions, respectively.1 This convention ensures that mirroring the diagram across its main diagonal will produce the Young tableau of another overpartition, more commonly known as conjugating the overpartition.

Variations of Andrews-Beck type congruences

“…Here, (4.9) implies [10, Corollary 1.5]. Second, do similar congruences and/or relations exist for total number of parts functions associated to other partitions statistics, for example, ranks of Durfee symbols [1], the first and second residual crank for overpartitions [7], the k-rank [12] or the M d -rank for overpartitions [17], [23]? Finally, a mock modular perspective (such as in [13], [15], [16] or [22]) which explains the occurrences of (1.1), (1.2), (1.4)-(1.7) and (4.3)-(4.16) would be most welcome.…”

Quasimodularity of the $k$th Residual Cranks

“…At this point in time, we are not aware of a suitable rank function to pair with the kth residual crank function for k > 2. Although the first author's M k -ranks [Mor19] generalize the rank functions studied by Bringmann, Lovejoy, and Osburn, these fail to produce the expected spt relations. Where we would expect…”

Inequalities for the dth residual crank moments of overpartitions

“…can be shown using the same argument as in Larsen, Rust, and Swisher [LRS14]. Here, N[d] k (n) is the kth positive moment of the M d -rank of overpartitions, which is due to the second author [Mor19]. These rank moments have the following generating series n≥0…”