A333901 - OEIS

A333901

Array read by antidiagonals: T(n,k) is the number of n X k nonnegative integer matrices with all column sums n and row sums k.

%I #11 Apr 21 2020 19:28:43

%S 1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,7,7,1,1,1,1,19,55,19,1,1,1,1,51,

%T 415,415,51,1,1,1,1,141,3391,10147,3391,141,1,1,1,1,393,28681,261331,

%U 261331,28681,393,1,1,1,1,1107,248137,7100821,22069251,7100821,248137,1107,1,1

%N Array read by antidiagonals: T(n,k) is the number of n X k nonnegative integer matrices with all column sums n and row sums k.

%C T(n,k) is the number of ordered factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity).

%H Andrew Howroyd, <a href="/A333901/b333901.txt">Table of n, a(n) for n = 0..405</a> (antidiagonals n=0..27)

%F T(n,k) = T(k,n).

%e Array begins:

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

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

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

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

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

%e 2 | 1 1 3 7 19 51 141 ...

%e 3 | 1 1 7 55 415 3391 28681 ...

%e 4 | 1 1 19 415 10147 261331 7100821 ...

%e 5 | 1 1 51 3391 261331 22069251 1985311701 ...

%e 6 | 1 1 141 28681 7100821 1985311701 602351808741 ...

%e ...

%e The T(3,2) = 7 matrices are:

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

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

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

%o (PARI)

%o T(n, k)={

%o local(M=Map(Mat([k, 1])));

%o my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));

%o my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(n, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, n, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));

%o for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])

%o }

%o for(n=0, 7, for(k=0, 7, print1(T(n,k), ", ")); print)

%Y Columns k=0..9 are A000012, A000012, A002426, A172743, A172816, A172868, A172904, A172931, A172947, A172961.

%Y Main diagonal is A110058.

%Y Cf. A257462, A257493.

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, Apr 18 2020