OFFSET
0,15
COMMENTS
T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column having at least two nonzero entries 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) = A307316(n).
EXAMPLE
Triangle begins:
1;
0;
0, 0;
0, 0;
0, 0, 1;
0, 0, 1;
0, 0, 5, 2;
0, 0, 6, 3;
0, 0, 16, 16, 5;
0, 0, 22, 44, 13;
0, 0, 45, 135, 82, 11;
0, 0, 64, 338, 301, 52;
0, 0, 119, 880, 1233, 382, 34;
0, 0, 171, 2024, 4090, 1936, 211;
...
The T(6,2) = 5 multiset partitions are:
{{1,1,1,2}, {1,2}},
{{1,1,2,2}, {1,2}},
{{1,1,2}, {1,1,2}},
{{1,1,2}, {1,2,2}},
{{1,2,3}, {1,2,3}}.
The corresponding T(6,2) = 5 matrices are:
[3 1] [2 2] [2 1] [2 1] [1 1 1]
[1 1] [1 1] [2 1] [1 2] [1 1 1]
The T(6,3) = 2 matrices are:
[1 1] [1 1 0]
[1 1] [1 0 1]
[1 1] [0 1 1]
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]))); eta(x + O(x*x^k))*(1 + x*Ser(K(q, t, k))) + x*(1-c)/(1-x) - 1}
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