A Stanley–Elder theorem on Cranks and Frobenius symbols

Abstract

The Stanley-Elder theorem asserts that the number of j's in the partitions of n is equal to the number of parts that appear at least j times in a given partition of n, summed over all partitions of n. In this paper, we prove that the number of partitions of n with crank > j equals to half the total number of j's in the Frobenius symbols for n.

Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions

“…The novelty of Theorem 1.10 is that it identifies the odd crank enumeration of partitions with those partitions into odd number of parts and self-conjugate partitions through Liouville's function λ. Following the work done in [6,18], Theorem 1.11 springs up rather organically. Here, we count Frobenius symbols with restrictions on the entries and equate them to the enumeration of number of partitions with no parts that equal the size of the Durfee square of that partition, two ideas in the theory of partitions that are very rarely correlated.…”

Combinatorial Perspectives on the Crank and Mex Partition Statistics

“…Andrews, Dastidar, and Morrill give an alternative analytic proof of [23,Theorem 8] in their [3,Theorem 2], stated in an equivalent form. They conclude their paper, Prior to the discoveries of this paper and [Hopkins, Sellers, Stanton], there was no reason to suspect that there would be any connection between Cranks and Frobenius symbols.…”

“…Given the Ferrers diagram of a partition, the top row of the Frobenius symbol gives the number of boxes to the right of the diagonal entries and the bottom row gives the number of boxes below the diagonal entries. In subsequent work, Andrews, Dastidar, and Morrill [3] have found additional connections between the crank and the Frobenius symbol and, at the end of their article, ask for combinatorial insight to these new relations. We believe the work in Section 4 helps answer that request.…”

“…There has been a surge of research on the mex and these related concepts. In addition to [3,24] mentioned above and discussed in more detail below, the interested reader will want to look at [7,8,10,11,14,15,25]. There are even very recent results [9,12,26] that build on a preprint version of this current work.…”