A254233 -id:A254233 - cp's OEIS frontend

A257462 Number A(n,k) of 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); 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, 4, 2, 1, 1, 1, 1, 10, 10, 3, 1, 1, 1, 1, 26, 70, 25, 3, 1, 1, 1, 1, 71, 566, 465, 49, 4, 1, 1, 1, 1, 197, 4781, 11131, 2505, 103, 4, 1, 1, 1, 1, 554, 41357, 297381, 190131, 12652, 184, 5, 1, 1, 1, 1, 1570, 364470, 8349223, 16669641, 2928876, 57232, 331, 5, 1, 1

Offset: 0

Alois P. Heinz, Apr 24 2015

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

			A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*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) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.
Square array A(n,k) begins:
  1, 1, 1,  1,    1,      1, ...
  1, 1, 1,  1,    1,      1, ...
  1, 1, 2,  4,   10,     26, ...
  1, 1, 2, 10,   70,    566, ...
  1, 1, 3, 25,  465,  11131, ...
  1, 1, 3, 49, 2505, 190131, ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals n=0..26)

Columns k=0+1, 2-5 give: A000012, A008619, A254233, A257114, A257518.
Rows n=0+1, 2-3 give: A000012, A257520, A333902.
Main diagonal gives A334286.
Cf. A257463, A333901 (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) minus {1}))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$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, Sum[If[d > i || PrimeOmega[d] != k, 0, b[n/d, d, k]], {d, Divisors[n] // Rest}]]; A[n_, k_] := Module[ {p = Product[Prime[i], {i, 1, k}]^n}, b[p, p, k]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A254243 Number of ways to partition the multiset consisting of 3 copies each of 1, 2, ..., n into n sets of size 3.

Original entry on oeis.org

1, 1, 2, 10, 93, 1417, 32152, 1016489, 42737945, 2307295021, 155607773014, 12823004639504, 1267907392540573, 148160916629902965, 20199662575448858212, 3177820001990224608763, 571395567211112572679633, 116448309072281063992943561, 26700057600529091443246943530

Offset: 0

Tatsuru Murai, Jan 27 2015

nonn

			a(1) = 1: 111.
a(2) = 2: 111|222 and 112|122.
a(3) = 10: 111|222|333, 111|223|233, 112|122|333, 112|123|233, 112|133|223, 113|122|233, 113|123|223, 113|133|222, 122|123|133, and 123|123|123.

Andrew Howroyd, Table of n, a(n) for n = 0..100
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. See page 40 (but there is a typo).
StackExchange, Number of Partitioning a deck with m cards in n types into n-element sets, January 2015.

Cf. A002135 (2 instead of 3), A254233 (n copies each of 1, 2, and 3).

Column k=3 of A257463.

Name and example edited by Danny Rorabaugh, Apr 22 2015
a(6)-a(10) from Alois P. Heinz, Apr 22 2015
Terms a(11) and beyond from Andrew Howroyd, Apr 18 2020

cp's OEIS Frontend

A257462 Number A(n,k) of 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); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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