OFFSET
0,8
COMMENTS
The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..209
FORMULA
EXAMPLE
Array begins:
===============================================================
n\k | 0 1 2 3 4 5 6
----+----------------------------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 1 1 2 4 8 16 32 ...
2 | 1 0 3 23 290 4298 79143 ...
3 | 1 0 0 184 17488 2780752 689187720 ...
4 | 1 0 0 840 771305 1496866413 5261551562405 ...
5 | 1 0 0 0 21770070 585897733896 30607728081550686 ...
6 | 1 0 0 0 328149360 161088785679360 ...
...
The A(2,2) = 3 matrices are:
[1 1] [1 0] [1 0]
[1 0] [1 1] [0 1]
[0 1] [0 1] [1 1]
PROG
(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 20 2020
STATUS
approved