A254243 -id:A254243 - cp's OEIS frontend

A257463 Number A(n,k) of factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 3, 10, 17, 1, 1, 1, 1, 3, 23, 93, 73, 1, 1, 1, 1, 4, 40, 465, 1417, 388, 1, 1, 1, 1, 4, 73, 1746, 19834, 32152, 2461, 1, 1, 1, 1, 5, 114, 5741, 190131, 1532489, 1016489, 18155, 1, 1

Offset: 0

Alois P. Heinz, Apr 24 2015

Also number of ways to partition the multiset consisting of k copies each of 1, 2, ..., n into n multisets of size k.

			A(4,2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4.
A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.
A(2,4) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,        1,         1, ...
  1, 1,   1,     1,       1,        1,         1, ...
  1, 1,   2,     2,       3,        3,         4, ...
  1, 1,   5,    10,      23,       40,        73, ...
  1, 1,  17,    93,     465,     1746,      5741, ...
  1, 1,  73,  1417,   19834,   190131,   1398547, ...
  1, 1, 388, 32152, 1532489, 43816115, 848597563, ...

Andrew Howroyd, Antidiagonals n = 0..27, flattened (antidiagonals 0..12 from Alois P. Heinz)
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. The array is on page 40.
Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets
Marko Riedel, Maple program to compute array from cycle indices

Columns k=0+1, 2-4 give: A000012, A002135, A254243, A268668.
Rows n=0+1, 2-5 give: A000012, A008619, A257464, A253259, A253263.
Main diagonal gives A334286.
Cf. A257462, A257493 (ordered factorizations).

with(numtheory):
b:= proc(n, i, k) option remember; `if`(n=1, 1,
      add(`if`(d>i or bigomega(d)<>k, 0,
      b(n/d, d, k)), d=divisors(n)))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, i_, k_] := b[n, i, k] = If[n==1, 1, DivisorSum[n, If[#>i || PrimeOmega[#] != k, 0, b[n/#, #, k]]&]];
A[n_, k_] := b[p = Product[Prime[i], {i, 1, n}]^k, p, k];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)

A254233 Number of ways to partition the multiset consisting of n copies each of 1, 2, and 3 into n sets of size 3.

Original entry on oeis.org

1, 1, 4, 10, 25, 49, 103, 184, 331, 554, 911, 1424, 2204, 3278, 4817, 6896, 9746, 13487, 18480, 24882, 33192, 43683, 56994, 73512, 94131, 119340, 150300, 187732, 233065, 287248, 352153, 428944, 519949, 626737, 752095, 897994, 1067924, 1264241, 1491155, 1751672

Offset: 0

Tatsuru Murai, Jan 27 2015

			For n = 2, the set {1,1,2,2,3,3} can be partitioned into two sets in four ways: {{112},{233}}, {{113},{223}}, {{122},{133}}, and {{123},{123}}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A002135, A254243.

Column k=3 of A257462.

G.f.: (x^12-x^11+x^10+3*x^9+5*x^8+x^7+4*x^6+x^5+5*x^4+3*x^3+x^2-x+1) / ((x^2+1)*(x^2-x+1)*(x^2+x+1)^3*(x+1)^4*(x-1)^8). - Alois P. Heinz, Apr 21 2015

Fixed definition and examples by Kellen Myers, Apr 21 2015

a(14)-a(39) from Alois P. Heinz, Apr 21 2015

A333899 Number of nonequivalent n X n binary matrices with 3 ones in every row and column up to permutation of rows.

Original entry on oeis.org

1, 0, 0, 1, 1, 22, 550, 16700, 703297, 38135272, 2584332084, 214050246975, 21269928579625, 2497135296996934, 341930210261762434, 54008226638062039798, 9746833030828647070625, 1993080976749664740373600, 458405276998431265342151848, 117814680468332544208002721773

Offset: 0

Andrew Howroyd, Apr 18 2020

nonn

Number of factorizations of m^3 into n factors, where m is a product of exactly n distinct primes and each factor is a product of 3 distinct primes.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=3 of A260340.

Cf. A254243.

cp's OEIS Frontend

A260339 Erroneous version of A254243.

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A257463 Number A(n,k) of factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A254233 Number of ways to partition the multiset consisting of n copies each of 1, 2, and 3 into n sets of size 3.

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A333899 Number of nonequivalent n X n binary matrices with 3 ones in every row and column up to permutation of rows.

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