A331160 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and rows in decreasing lexicographic order.
1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 2, 1, 0, 1, 27, 15, 2, 1, 0, 1, 266, 317, 44, 3, 1, 0, 1, 3599, 12586, 2763, 120, 4, 1, 0, 1, 62941, 803764, 390399, 21006, 319, 5, 1, 0, 1, 1372117, 75603729, 103678954, 10074052, 147296, 804, 6, 1
Offset: 0
Examples
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 1 4 27 266 3599 62941 ...
3 | 1 2 15 317 12586 803764 75603729 ...
4 | 1 2 44 2763 390399 103678954 46278915417 ...
5 | 1 3 120 21006 10074052 10679934500 21806685647346 ...
6 | 1 4 319 147296 232165926 956594630508 8717423133548684 ...
7 | 1 5 804 967829 4903530137 76812482919237 ...
...
The A(2,2) = 4 matrices are:
[2 1] [2 0] [1 2] [1 1]
[0 1] [0 2] [1 0] [1 0]
[0 1]
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..209
Crossrefs
Programs
-
PARI
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)*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)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}
Comments