A257462 -id:A257462 - cp's OEIS frontend

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.

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, 415, 415, 51, 1, 1, 1, 1, 141, 3391, 10147, 3391, 141, 1, 1, 1, 1, 393, 28681, 261331, 261331, 28681, 393, 1, 1, 1, 1, 1107, 248137, 7100821, 22069251, 7100821, 248137, 1107, 1, 1

Offset: 0

Andrew Howroyd, Apr 18 2020

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).

			Array begins:
=======================================================
n\k | 0 1   2     3       4          5            6
----+--------------------------------------------------
  0 | 1 1   1     1       1          1            1 ...
  1 | 1 1   1     1       1          1            1 ...
  2 | 1 1   3     7      19         51          141 ...
  3 | 1 1   7    55     415       3391        28681 ...
  4 | 1 1  19   415   10147     261331      7100821 ...
  5 | 1 1  51  3391  261331   22069251   1985311701 ...
  6 | 1 1 141 28681 7100821 1985311701 602351808741 ...
  ...
The T(3,2) = 7 matrices are:
  [1 1]  [1 1]  [1 1]  [2 0]  [2 0]  [0 2]  [0 2]
  [1 1]  [2 0]  [0 2]  [1 1]  [0 2]  [1 1]  [2 0]
  [1 1]  [0 2]  [2 0]  [0 2]  [1 1]  [2 0]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..405 (antidiagonals n=0..27)

Columns k=0..9 are A000012, A000012, A002426, A172743, A172816, A172868, A172904, A172931, A172947, A172961.
Main diagonal is A110058.
Cf. A257462, A257493.

T(n, k)={
  local(M=Map(Mat([k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  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)))));
  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])
}
for(n=0, 7, for(k=0, 7, print1(T(n,k), ", ")); print)

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

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.

Original entry on oeis.org

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 *)

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946

Offset: 0

Gus Wiseman, Oct 12 2018

nonn

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022

			The a(9) = 26 multiset partitions:
  {{9}}
  {{1,8}}
  {{2,7}}
  {{3,6}}
  {{4,5}}
  {{1,2,6}}
  {{1,3,5}}
  {{1},{8}}
  {{2,3,4}}
  {{2},{7}}
  {{3,3,3}}
  {{3},{6}}
  {{4},{5}}
  {{1},{2},{6}}
  {{1},{3},{5}}
  {{2},{3},{4}}
  {{3},{3},{3}}
  {{1,1,1,2,2,2}}
  {{1,1,1},{2,2,2}}
  {{1,1,2},{1,2,2}}
  {{1,1},{1,2},{2,2}}
  {{1,2},{1,2},{1,2}}
  {{1,1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,1,1},{1,1,1}}
  {{1},{1},{1},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..150
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]

Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022

A257520 Number of factorizations of m^2 into 2 factors, where m is a product of exactly n distinct primes and each factor is a product of n primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 71, 197, 554, 1570, 4477, 12827, 36895, 106471, 308114, 893804, 2598314, 7567466, 22076405, 64498427, 188689685, 552675365, 1620567764, 4756614062, 13974168191, 41088418151, 120906613076, 356035078102, 1049120176954, 3093337815410

Offset: 0

Alois P. Heinz, Apr 27 2015

nonn

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

			a(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.

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

Row n=2 of A257462.

Cf. A002426, A097861.

a:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
      ((3*n^2-7*n+3)*a(n-1) +(n-1)*(n-3)*a(n-2)
       -3*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))
    end:
seq(a(n), n=0..40);

G.f.: (1/sqrt((1+x)*(1-3*x))+1/(1-x))/2.
E.g.f.: exp(x)*(1+BesselI(0,2*x))/2.
a(n) = ((3*n^2-7*n+3)*a(n-1) +(n-1)*(n-3)*a(n-2) -3*(n-1)*(n-2)*a(n-3)) / (n*(n-2)) for n>2, a(0) = a(1) = 1, a(2) = 2.
a(n) = (A002426(n)+1)/2.
a(n) = A097861(n)+1.

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

A333902 Number of nonequivalent 3 X n nonnegative integer matrices with all column sums 3 and row sums n up to permutation of rows.

Original entry on oeis.org

1, 1, 2, 10, 70, 566, 4781, 41357, 364470, 3257830, 29450557, 268701797, 2470513849, 22862694505, 212758007450, 1989477010250, 18682005254390, 176085344355510, 1665167298000005, 15793406377949405, 150192477267201945, 1431741484334863145, 13678227386467491410

Offset: 0

Andrew Howroyd, Apr 18 2020

nonn

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

			The a(2) = 2 matrices are:
  [1 1]  [0 2]
  [1 1]  [1 1]
  [1 1]  [2 0]

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

Row n=3 of A257462.

A334286 Number of factorizations of m^n into n factors, where m is a product of exactly n distinct primes and each factor is a product of n primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 10, 465, 190131, 848597563, 43025433375905, 26004966055138634525, 194173310204064149748222455, 18434259996904142171888712495703426, 22778257480946919793779826285286813732062310, 373444566958856976964193391832469245535883039838631492

Offset: 0

Alois P. Heinz, Apr 21 2020

nonn

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

			a(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.

Main diagonal of A257462 and of A257463.

a(n) = A257462(n,n) = A257463(n,n).

More terms from Andrew Howroyd, Apr 21 2020

A257114 Number of factorizations of m^n into n factors, where m is a product of exactly 4 distinct primes and each factor is a product of 4 primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 10, 70, 465, 2505, 12652, 57232, 240481, 936785, 3428138, 11817866, 38676949, 120577553, 359800464, 1030830032, 2845200663, 7584911479, 19580001382, 49046743566, 119457712491, 283383330899, 655832583316, 1482829086428, 3279794012205, 7105246435381

Offset: 0

Alois P. Heinz, Apr 24 2015

nonn

			a(2) = 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.

Column k=4 of A257462.

A257518 Number of factorizations of m^n into n factors, where m is a product of exactly 5 distinct primes and each factor is a product of 5 primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 26, 566, 11131, 190131, 2928876, 40757006, 518890101, 6083666731, 66157864251, 671143606086, 6384844387411, 57220640955261, 485038299365181, 3902760531727911, 29904009021483406, 218823691308461156, 1533172142451741421, 10309493626479157551

Offset: 0

Alois P. Heinz, Apr 27 2015

nonn

			a(2) = 26: (2*3*5*7*11)^2 = 5336100 = 2310*2310 = 2420*2205 = 2450*2178 = 2475*2156 = 3234*1650 = 3388*1575 = 3465*1540 = 3630*1470 = 3675*1452 = 3850*1386 = 4851*1100 = 5082*1050 = 5390*990 = 5445*980 = 5775*924 = 6050*882 = 7623*700 = 8085*660 = 8470*630 = 9075*588 = 11858*450 = 12705*420 = 13475*396 = 17787*300 = 21175*252 = 29645*180.

Column k=5 of A257462.

cp's OEIS Frontend

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.

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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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A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

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A257520 Number of factorizations of m^2 into 2 factors, where m is a product of exactly n distinct primes and each factor is a product of n primes (counted with multiplicity).

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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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A333902 Number of nonequivalent 3 X n nonnegative integer matrices with all column sums 3 and row sums n up to permutation of rows.

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A334286 Number of factorizations of m^n into n factors, where m is a product of exactly n distinct primes and each factor is a product of n primes (counted with multiplicity).

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A257114 Number of factorizations of m^n into n factors, where m is a product of exactly 4 distinct primes and each factor is a product of 4 primes (counted with multiplicity).

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A257518 Number of factorizations of m^n into n factors, where m is a product of exactly 5 distinct primes and each factor is a product of 5 primes (counted with multiplicity).

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