Congruences for Partition Quadruples with t -Cores
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Congruences for Partition Quadruples with t -Cores M. S. Mahadeva Naika1 · S. Shivaprasada Nayaka1 Received: 30 May 2018 / Accepted: 3 October 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019
Abstract Let Ct (n) denote the number of partition quadruples of n with t-cores for t = 3, 5, 7, 25. We establish some Ramanujan type congruences modulo 5, 7, 8 for Ct (n). For example, n ≥ 0, we have C5 (5n + 4) ≡ 0 (mod 5), C7 (7n + 6) ≡ 0 (mod 7), C3 (16n + 14) ≡ 0 (mod 8). Keywords Congruences · Partition quadruples · t-core partition Mathematics Subject Classification (2010) 11P81 · 11P83
1 Introduction A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. The Ferrers-Young diagram of the partition λ of n is obtained by arranging n nodes in k left aligned rows so that the ith row has λi nodes. The nodes are labeled by row and column coordinates as one would label the entries of a matrix. Let λj denote the number of nodes in column j . The hook number H (i, j ) of the (i, j ) node is defined as the number of nodes directly below and to the right of the node including the node itself, i.e., H (i, j ) = λi + λj − j − i + 1. A t-core is a partition with no hook number that are divisible by t.
M. S. Mahadeva Naika
[email protected] S. Shivaprasada Nayaka [email protected] 1
Department of Mathematics, Bangalore University, Central College Campus, Bengaluru 560 001, Karnataka, India
M.S.M. Naika, S.S. Nayaka
For example, the Ferrers-Young diagram of the partition λ = (5, 3, 2) of 10 is
The nodes (1, 1), (1, 2), (1, 3), (1, 4), (1,5), (2, 1), (2, 2), (2, 3), (3, 1), and (3, 2) have hook numbers 7, 6, 4, 2, 1, 4, 3, 1, 2, and 1, respectively. Therefore, λ is a t-core partition for t = 5 and for all t ≥ 8. Let at (n) be the number of partitions of n that are t-cores. Then, its generating function is given by [4, Eq. (2.1)] ∞ (q t ; q t )t∞ at (n)q n = . (q; q)∞ n=0 Ramanujan’s three famous congruences of p(n) are as follows: p(5n + 4) ≡ 0
(mod 5),
p(7n + 5) ≡ 0
(mod 7),
p(11n + 6) ≡ 0
(mod 11).
In [5, 6], Hischhorn and Sellers have studied the 4-core partition (i.e., a4 (n)) and established some infinite families of arithmetic relations for a4 (n). Baruah and Nath [1] have proved some more infinite families of arithmetic identities for a4 (n). A bipartition of n is a pair of partitions (λ1 , λ2 ) such that the sum of all parts of λ1 and λ2 equals n. A bipartition with t-core of n is a bipartition (λ1 , λ2 ) of n such that λ1 and λ2 are both t-cores. Let At (n) denote the number of bipartitions with t-cores of n. The generating function for At (n) is given by ∞
At (n)q n =
n=0
(q t ; q t )2t ∞ . (q; q)2∞
Recently, Lin [8] has established some congruence and infinite families for A3 (n). In [2], Baruah and Nath have found three infinite families of A3 (n). A partition (λ1 , λ2 , . . . , λk ) of a positive integer n is a k-tuple of partitions such that the sum of all th
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