A322487 -id:A322487 - cp's OEIS frontend

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

Alois P. Heinz, Nov 26 2012

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

			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, ...

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

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.

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

A358721 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 2, 2, 2, ..., n, n, n] into k nonempty submultisets, for 1 <= k <= 3n.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 11, 8, 3, 1, 1, 31, 139, 219, 175, 86, 28, 6, 1, 1, 127, 1547, 5321, 8004, 6687, 3579, 1329, 359, 71, 10, 1, 1, 511, 16171, 118605, 333887, 472784, 398771, 223700, 89640, 26853, 6171, 1100, 150, 15, 1, 1, 2047, 164651, 2511653, 13045458, 31207637, 41429946, 34621129, 19882236, 8342411, 2668319, 669446, 134075, 21591, 2785, 281, 21, 1

Offset: 0

Marko Riedel, Nov 28 2022

A generalization of ordinary Stirling set numbers to multisets that contain some m instances each of n elements, here we have m=3.

			The triangular array starts:
[0]: 1,
[1]: 1,   1,    1;
[2]: 1,   7,   11,    8,    3,    1;
[3]: 1,  31,  139,  219,  175,   86,   28,    6,   1;
[4]: 1, 127, 1547, 5321, 8004, 6687, 3579, 1329, 359, 71, 10, 1;

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Number of ways to partition a multiset into k non-empty multisets, Mathematics Stack Exchange.
Marko Riedel, Maple code for sequence by plain enumeration, the Polya Enumeration Theorem, and Power Group Enumeration

Cf. A008277, A358710, A358722, A322487 (row sums).

A331196 Number of nonnegative integer matrices with n distinct columns and any number of nonzero rows with each column sum being 3 and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 28, 599, 23243, 1440532, 131530132, 16720208200, 2837752812927, 622570020892599, 172077041175850521, 58679982298020226625, 24262822372018694983540, 11986886218243164848742812, 6987708088810202717378639087, 4754544525981425409034078100189

Offset: 0

Andrew Howroyd, Jan 11 2020

nonn

The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows.

			The a(2) = 28 matrices include 6 with 2 rows, 10 with 3 rows, 8 with 4 rows, 3 with 5 rows and 1 with 6 rows. The 16 with 2 or 3 rows are:
   [3 2]  [3 1]  [3 0]  [2 3]  [2 1]  [2 0]  [3 1]  [3 0]
   [0 1]  [0 2]  [0 3]  [1 0]  [1 2]  [1 3]  [0 1]  [0 2]
                                             [0 1]  [0 1]
.
   [2 2]  [2 1]  [2 1]  [2 0]  [2 0]  [2 0]  [1 3]  [1 2]
   [1 0]  [1 1]  [1 0]  [1 2]  [1 1]  [1 0]  [1 0]  [1 1]
   [0 1]  [0 1]  [0 2]  [0 1]  [0 2]  [0 3]  [1 0]  [1 0]

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

Row n=3 of A331161.

Cf. A322487.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A322487(k).

A319591 Number of nonnegative integer matrices with n columns and any number of nonzero distinct rows with every column summing to 3 up to permutation of rows.

Original entry on oeis.org

1, 2, 17, 364, 14595, 937776, 88507276, 11584785137, 2017129470049, 452573312572094, 127585778625167901, 44275881599081757633, 18594652294164085489646, 9315786786179883210141889, 5499383628157822564248546214, 3784760890972848935690646794792

Offset: 0

Andrew Howroyd, Dec 16 2018

nonn

Also, the number of factorizations of m^3 into distinct factors where m is a product of n distinct primes.

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

Row n=3 of A219585.

Cf. A322487.

A358781 Number of multiset partitions of [1,1,1,1,2,2,2,2,...,n,n,n,n] into nonempty multisets.

Original entry on oeis.org

1, 5, 109, 6721, 911838, 231575143, 99003074679, 66106443797808, 65197274052335504, 90954424202936106523, 173398227073956386079670, 439196881673194611574668282, 1443741072199958276777413001395

Offset: 0

Marko Riedel, Nov 29 2022

nonn

Generalization of Bell numbers to multiset partitions with m instances each of n different elements, here m=4.

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Number of ways to partition a multiset into k non-empty multisets, Mathematics Stack Exchange.

Cf. A358722, A020555, A322487.

Row n=4 of A219727.

cp's OEIS Frontend

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.

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A358721 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 2, 2, 2, ..., n, n, n] into k nonempty submultisets, for 1 <= k <= 3n.

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A331196 Number of nonnegative integer matrices with n distinct columns and any number of nonzero rows with each column sum being 3 and rows in nonincreasing lexicographic order.

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A319591 Number of nonnegative integer matrices with n columns and any number of nonzero distinct rows with every column summing to 3 up to permutation of rows.

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A358781 Number of multiset partitions of [1,1,1,1,2,2,2,2,...,n,n,n,n] into nonempty multisets.

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