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A219585
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Number A(n,k) of k-partite partitions of {n}^k into distinct k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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14
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 2, 1, 1, 15, 40, 17, 2, 1, 1, 52, 457, 364, 46, 3, 1, 1, 203, 6995, 14595, 2897, 123, 4, 1, 1, 877, 136771, 937776, 407287, 21369, 323, 5, 1, 1, 4140, 3299218, 88507276, 107652681, 10200931, 148257, 809, 6, 1
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OFFSET
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0,8
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COMMENTS
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A(n,k) is the number of factorizations of m^n into distinct factors where m is a product of k distinct primes. A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into distinct factors: 36, 3*12, 4*9, 2*18, 2*3*6.
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LINKS
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FORMULA
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A(n,k) = [(Product_{j=1..k} x_j)^n] 1/2 * Product_{i_1,...,i_k>=0} (1+Product_{j=1..k} x_j^i_j).
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EXAMPLE
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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(3,2) = 17: [(3,3)], [(3,0),(0,3)], [(3,2),(0,1)], [(2,3),(1,0)], [(3,1),(0,2)], [(2,2),(1,1)], [(1,3),(2,0)], [(2,1),(1,2)], [(2,1),(1,1),(0,1)], [(3,0),(0,2),(0,1)], [(2,2),(1,0),(0,1)], [(2,1),(0,2),(1,0)], [(1,2),(2,0),(0,1)], [(1,2),(1,1),(1,0)], [(0,3),(2,0),(1,0)], [(2,0),(1,1),(0,2)], [(2,0),(0,2),(1,0),(0,1)].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 5, 15, 52, 203, ...
1, 1, 5, 40, 457, 6995, 136771, ...
1, 2, 17, 364, 14595, 937776, 88507276, ...
1, 2, 46, 2897, 407287, 107652681, ...
1, 3, 123, 21369, 10200931, 10781201973, ...
1, 4, 323, 148257, 233051939, ...
1, 5, 809, 970246, 4909342744, ...
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MATHEMATICA
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f[n_, k_] := f[n, k] = 1/2 Product[Sum[O[x[j]]^(n+1), {j, 1, k}]+1+ Product[x[j]^i[j], {j, 1, k}], Evaluate[Sequence @@ Table[{i[j], 0, n}, {j, 1, k}]]];
a[0, _] = a[_, 0] = 1; a[n_, k_] := SeriesCoefficient[f[n, k], Sequence @@ Table[{x[j], 0, n}, {j, 1, k}]];
Table[Print[a[n-k, k]]; a[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013, updated Sep 16 2019 *)
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PROG
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(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+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 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)} \\ Andrew Howroyd, Dec 16 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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