OFFSET
0,13
COMMENTS
The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
A(n,k) is the number of k-block n-uniform T_0 set systems without isolated vertices.
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 0 0 0 0 0 ...
1 | 1 1 1 1 1 1 1 ...
2 | 1 0 3 17 230 3264 60338 ...
3 | 1 0 0 184 16936 2711904 675457000 ...
4 | 1 0 0 840 768785 1493786233 5254074934990 ...
5 | 1 0 0 0 21770070 585810653616 30604798810581906 ...
6 | 1 0 0 0 328149360 161087473081920 ...
...
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)/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, k<=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