OFFSET
1,3
COMMENTS
T(n,k) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
EXAMPLE
Triangle begins:
1;
1, 2;
1, 2, 3;
1, 5, 5, 5;
1, 5, 12, 9, 7;
1, 9, 23, 29, 17, 11;
1, 9, 39, 62, 57, 28, 15;
1, 14, 63, 147, 154, 110, 47, 22;
...
The T(4,3) = 5 matrices are:
[1 0 0] [1 0 0] [1 1 0] [1 1 1] [1 1 0]
[1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 1]
[0 1 0] [0 1 1] [0 0 1]
[0 0 1]
The T(4,3) = 5 the set multipartitions are:
{{1,2}, {3}, {4}},
{{1,2}, {3}, {3}},
{{1,2}, {1}, {3}},
{{1,2}, {1}, {1}},
{{1,2}, {1}, {2}}.
PROG
(PARI) \\ See A321609 for definition of M.
T(n, k)={M(k, n, n) - M(k-1, n, n)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print)
(PARI) \\ Faster version.
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, n)={prod(j=1, #q, (1+x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
G(m, n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
A(n, m=n)={my(p=sum(k=0, m, G(k, n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jul 03 2020
STATUS
approved