A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 9, 3, 1, 1, 15, 66, 31, 5, 1, 1, 52, 712, 686, 109, 7, 1, 1, 203, 10457, 27036, 6721, 339, 11, 1, 1, 877, 198091, 1688360, 911838, 58616, 1043, 15, 1, 1, 4140, 4659138, 154703688, 231575143, 26908756, 476781, 2998, 22, 1

Offset

0,8

Comments

A(n,k) is the number of factorizations of m^n where m is a product of k distinct primes. A(2,2) = 9: (2*3)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.

A(n,k) is the number of (n*k) X k matrices with nonnegative integer entries and column sums n up to permutation of rows. - Andrew Howroyd, Dec 10 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 0..209

Example

A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
A(2,2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].
Square array A(n,k) begins:
  1,   1,    1,      1,        1,         1,         1,       1, ...
  1,   1,    2,      5,       15,        52,       203,     877, ...
  1,   2,    9,     66,      712,     10457,    198091, 4659138, ...
  1,   3,   31,    686,    27036,   1688360, 154703688, ...
  1,   5,  109,   6721,   911838, 231575143, ...
  1,   7,  339,  58616, 26908756, ...
  1,  11, 1043, 476781, ...
  1,  15, 2998, ...

Prog

(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]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
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, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 11 2018

Crossrefs

Columns k=0..3 give: A000012, A000041, A002774, A219678.

Rows n=0..4 give: A000012, A000110, A020555, A322487, A358781.

Main diagonal gives A322488.

Cf. A188392, A219585 (partitions of {n}^k into distinct k-tuples), A256384, A318284, A318951.

Sequence in context: A129104 A232648 A295690 * A177694 A092450 A279629

Adjacent sequences: A219724 A219725 A219726 * A219728 A219729 A219730

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 26 2012

Status

approved