The spt-Function of Andrews

William Y. C. Chen

  Abstract:   The spt-function spt(n) was introduced by Andrews as the weighted counting of partitions of n with respect to the number of occurrences of the smallest part. Andrews showed that spt(5n+4)≡ 0 (mod 5), spt(7n+5) ≡ 0 (mod 7) and spt(13n+6) ≡ 0 (mod 13). Since then, congruences of spt(n) have been extensively studied. Folsom and Ono obtained congruences of spt(n) mod 2 and 3. They also showed that the generating function of spt(n) mod 3 is related to a weight 3/2 Hecke eigenform with Nebentypus. Combinatorial interpretations of congruences of spt(n) mod 5 and 7 have been found by Andrews, Garvan and Liang by introducing the spt-crank of a vector partition. Chen, Ji and Zang showed that the set of partitions counted by spt(5n+4) (or spt(7n+5)) can be divided into five (or seven) equinumerous classes according to the spt-crank of a doubly marked partition. Let NS(m, n) denote the net number of S-partitions of n with spt-crank m. Andrews, Dyson and Rhoades conjectured that {NS(m, n)}m is unimodal for any n. Chen, Ji and Zang gave a constructive proof of this conjecture. In this survey, we summarize developments on congruence properties of spt(n) established by Andrews, Bringmann, Folsom, Garvan, Lovejoy and Ono et al., as well as their combinatorial interpretations. Generalizations and variations of the spt-function are also discussed. We further give an overview of asymptotic formulas of spt(n) obtained by Ahlgren, Andersen and Rhoades et al. We conclude with some conjectures on inequalities on spt(n), which are reminiscent of inequalities on p(n) due to DeSalvo and Pak, and Bessenrodt and Ono. Furthermore, we observe that, beyond the log-concavity, p(n) and spt(n) satisfy higher order inequalities based on polynomials arising in the invariant theory of binary forms. In particular, we conjecture that the higher order Turan inequality 4(an2 - an-1 an+1)(an+12 - an an+2) - (an an+1 - an-1 an+2)2 >0 holds for p(n) when n ≥ 95 and for spt(n) when n ≥ 108.

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