Generalisations of Capparelli's and Primc's identities, I: Coloured Frobenius partitions and combinatorial proofs

New partition identities from \(C^{(1)}_\ell\)-modules

“…there is 7 (0, 1, 0)admissible partitions of 6. The Python code 21AAIC.py is just a bit faster and more polished version of this code and gives for the number of all (0, 1, 0)admissible partitions of n ≤ N = 6 the list [3,3], [4,3], [5,5], [6,7]] Of course, this example would have been much shorter if we simply wrote "by hand" all (0, 1, 0)-admissible partitions of n ≤ 6, as we did for all (2, 0, 0)admissible partitions of n ≤ 8 in Example 3.1.…”

“…Here we give the Python code which is in part explained in Example 5.3 where diagonals in the array (3.3) became the rows in the matrix (5.12). [1, 1], [2,1], [3,1], [4,2], [5,2], [6,3], [7,3], [8,4], [9,5], [10,6], [11,7], [12,9], [13,10], [14,12], [15,14], [16,17], [17,19], [18,23], [19,26], [20,31]; highest_weight = [1, 0] [1, 0], [2,1], [3,1], [4,1], [5,1], [6,2], [7,2], [8,3]...…”

“…Recently several identities in the style of the Rogers-Ramanujan identities related to the representation theory of affine Lie algebras have appeared, let us mention only [13] and [5]. On the other side, some parts of representation theory lead to Rogers-Ramanujan type colored partition identities, let us mention only [7] and [18] for this vein of research.…”