A new congruence modulo \(25\) for \(1\)-shell totally symmetric plane partitions

E. X. W. Xia

Abstract


For any positive integer \(n\), let \(f(n)\) denote the number of 1-shell totally symmetric plane partitions of \(n\). Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.\(90\) (2014), 37–46] proved a number of congruences satisfied by \(f(n)\). In particular, Hirschhorn and Sellers proved that \(f(10n+5)≡0 (mod 5)\). In this paper, we establish the generating function of \(f(30n+25)\) and prove that \(f(250n+125)≡0 (mod 25)\).
DOI:

10.1017/S0004972714000768

Keywords


congruence, $1$-shell totally symmetric plane partitions, TSPP.



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