A331161 - OEIS

A331161

Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and rows in nonincreasing lexicographic order.

%I #10 Jan 24 2020 15:52:27

%S 1,1,1,0,1,1,0,1,2,1,0,1,7,3,1,0,1,43,28,5,1,0,1,403,599,104,7,1,0,1,

%T 5245,23243,6404,332,11,1,0,1,89132,1440532,872681,57613,1032,15,1,0,

%U 1,1898630,131530132,222686668,26560747,473674,2983,22,1

%N Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and rows in nonincreasing lexicographic order.

%C The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows

%H Andrew Howroyd, <a href="/A331161/b331161.txt">Table of n, a(n) for n = 0..209</a>

%F A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219727(n, j).

%F A330158(n) = Sum_{d|n} A(n/d, d).

%e Array begins:

%e ====================================================================

%e n\k | 0 1 2 3 4 5 6

%e ----+---------------------------------------------------------------

%e 0 | 1 1 0 0 0 0 0 ...

%e 1 | 1 1 1 1 1 1 1 ...

%e 2 | 1 2 7 43 403 5245 89132 ...

%e 3 | 1 3 28 599 23243 1440532 131530132 ...

%e 4 | 1 5 104 6404 872681 222686668 95605470805 ...

%e 5 | 1 7 332 57613 26560747 26852940027 52296207431182 ...

%e 6 | 1 11 1032 473674 712725249 2776638423133 ...

%e 7 | 1 15 2983 3599384 17328777789 ...

%e ...

%e The A(2,2) = 7 matrices are:

%e [2 1] [2 0] [1 2] [1 1] [2 0] [1 0] [1 0]

%e [0 1] [0 2] [1 0] [1 0] [0 1] [1 0] [1 0]

%e [0 1] [0 1] [0 2] [0 1]

%e [0 1]

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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]!)}

%o T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, k<=1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))}

%Y Rows n=1..3 are A000012, A014501, A331196.

%Y Columns k=0..2 are A000012, A000041, A331197.

%Y Cf. A219727, A330158, A331039, A331126, A331160, A331315.

%K nonn,tabl

%O 0,9

%A _Andrew Howroyd_, Jan 10 2020