OFFSET
0,8
COMMENTS
T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column sum at least two up to permutation of rows and columns.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..675 (rows 0..50)
FORMULA
T(2*n, n) = A050535(n).
EXAMPLE
Triangle begins:
1;
0;
0, 1;
0, 1;
0, 2, 3;
0, 2, 4;
0, 4, 15, 8;
0, 4, 24, 19;
0, 7, 60, 79, 23;
0, 8, 101, 213, 84;
0, 12, 210, 615, 424, 66;
0, 14, 357, 1523, 1533, 363;
0, 21, 679, 3851, 5580, 2217, 212;
0, 24, 1142, 8963, 17836, 10379, 1575;
...
The T(5,1) = 2 multiset partitions are:
{{1,1,1,1,1}},
{{1,1,1,2,2}}.
The corresponding T(5,1) = 2 matrices are:
[5] [3 2].
The T(5,2) = 4 matrices are:
[3] [3 0] [2 1] [2 1]
[2] [0 2] [1 1] [0 2],
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); (1 - x)*x*Ser(K(q, t, k)) - c*x}
G(n, y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
T(n)={my(v=Vec(G(n, 'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Jan 28 2024
STATUS
approved