Conjectural congruences for k-elongated plane partition functions modulo powers of 5

Establish the conjectured congruences, for all integers c,n≥0, d_{125c+58}(25n+16)≡0 (mod 125), d_{125c+83}(125n+41)≡d_{125c+83}(125n+91)≡0 (mod 125), d_{125c+100}(125n+124)≡0 (mod 125), d_{125c+5}(125n+69)≡d_{125c+5}(125n+119)≡0 (mod 125), d_{125c+30}(125n+69)≡0 (mod 125), d_{125c+60}(125n+14)≡d_{125c+60}(125n+64)≡d_{125c+60}(125n+89)≡d_{125c+60}(125n+114)≡0 (mod 125), d_{125c+58}(125n+91)≡0 (mod 625), and d_{125c+58}(125n+66)≡d_{125c+58}(125n+116)≡0 (mod 3125), thereby proving all congruences stated in Conjecture 7.1 for the k-elongated plane partition function d_k(n).

Background

The paper studies the k-elongated plane partition function d_k(n), whose generating function is f_2^k/f_1^{3k+1}, and proves several infinite families of congruences modulo 5, 25, and 125 using elementary q-series manipulations and 5-dissections. The authors explicitly state that the list of known congruences is not exhaustive and present additional congruences suggested by numerical calculations in Mathematica.

Conjecture 7.1 proposes eight further families. Six families concern congruences modulo 125, one strengthens a family to modulus 625, and one gives congruences modulo 3125. The paper notes that particular cases at c=0 may be approachable using Radu’s algorithm or the RaduRK package, but that a complete elementary proof of the conjecture remains to be supplied.