A note on the rank parity function. (English) Zbl 1228.11158
The rank of a partition is defined as the largest part minus the number of parts. Let \(S(n)\) denote the number of partitions into distinct parts with even rank minus those with odd rank. In the paper under review the author proves the following identity for positive even integers \(n\):
\[
S\left(\frac{n}{2}\right)=\sum_{\substack{ r\geq 0,\, |j|\leq r, \\ 24n+2=3(4r+1)^2-(6j+1)^2 }} (-1)^{r+j}-\sum_{\substack{ r\geq 0,\, |j|\leq r, \\ 24n+2=3(4r+3)^2-(6j+1)^2 }} (-1)^{r+j}.
\]
The author also establishes some results for the q-series related with the generating functions of \(S_E(n)\) and \(S_O(n)\), where \(S_E(n)\) (resp. \(S_O(n)\)) is the number of partitions of \(n\) into distinct parts with even rank minus the number of them with odd rank where the number of parts is even (resp. odd). The proofs of these results are achieved using two new Bailey pairs.
Reviewer: Ricardo Mamede (Coimbra)
MSC:
| 11P81 | Elementary theory of partitions |
| 05A15 | Exact enumeration problems, generating functions |
| 05A17 | Combinatorial aspects of partitions of integers |
| 11F27 | Theta series; Weil representation; theta correspondences |
Online Encyclopedia of Integer Sequences:
Expansion of Ramanujan’s function R(x) = 1 + Sum_{n >= 1} { x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.Expansion of Sum_{k>0} (-1)^(k+1) q^(k^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2k-1))).
G.f. Sum_{n >= 0} q^(4*n^2)/(q;q^2)_{2n}.
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