login
Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.
10

%I #10 Feb 01 2020 22:41:05

%S 1,1,1,1,1,1,1,2,2,1,1,3,7,3,1,1,5,23,21,5,1,1,7,79,162,66,7,1,1,11,

%T 274,1636,1338,192,11,1,1,15,1003,19977,43686,10585,565,15,1,1,22,

%U 3763,298416,2142277,1178221,82694,1579,22,1,1,30,14723,5300296,149056260,232984145,30370346,612700,4348,30,1

%N Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

%C A(n,k) is the number of non-isomorphic multiset partitions (multisets of multisets) with k parts each of size n.

%H Andrew Howroyd, <a href="/A331485/b331485.txt">Table of n, a(n) for n = 0..152</a>

%F A306017(n) = Sum_{d|n} A(n/d, d).

%e Array begins:

%e ============================================================

%e n\k | 0 1 2 3 4 5 6

%e ----+-------------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 ...

%e 1 | 1 1 2 3 5 7 11 ...

%e 2 | 1 2 7 23 79 274 1003 ...

%e 3 | 1 3 21 162 1636 19977 298416 ...

%e 4 | 1 5 66 1338 43686 2142277 149056260 ...

%e 5 | 1 7 192 10585 1178221 232984145 74676589469 ...

%e 6 | 1 11 565 82694 30370346 23412296767 33463656939910 ...

%e ...

%e The A(2,2) = 7 matrices are:

%e [1 0] [2 0] [1 1] [2 1] [2 0] [1 1] [2 2]

%e [1 0] [0 1] [1 0] [0 1] [0 2] [1 1]

%e [0 1] [0 1] [0 1]

%e [0 1]

%o (PARI) \\ See A318951 for RowSumMats

%o T(n, k)={RowSumMats(k, n*k, n)}

%o { for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }

%Y Rows n=0..4 are A000012, A000041, A007717, A058194, A331721.

%Y Columns k=0..3 are A000012, A000041, A331722, A331723.

%Y Cf. A219727, A306017, A316674, A318951, A331315, A331461.

%K nonn,tabl

%O 0,8

%A _Andrew Howroyd_, Jan 18 2020