A219727 -id:A219727 - cp's OEIS frontend

A330964 Array read by antidiagonals: A(n,k) is the number of sets of nonempty subsets of a k-element set where each element appears in at most n subsets.

1, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 15, 8, 2, 1, 1, 52, 59, 8, 2, 1, 1, 203, 652, 109, 8, 2, 1, 1, 877, 9736, 3623, 128, 8, 2, 1, 1, 4140, 186478, 200522, 11087, 128, 8, 2, 1, 1, 21147, 4421018, 16514461, 2232875, 21380, 128, 8, 2, 1, 1, 115975, 126317785, 1912959395, 775098224, 15312665, 29228, 128, 8, 2, 1

Offset: 0

Andrew Howroyd, Jan 04 2020

A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with rows in decreasing order and at most n ones in every column.

			Array begins:
==================================================================
n\k | 0 1 2   3     4         5             6                7
----+-------------------------------------------------------------
  0 | 1 1 1   1     1         1             1                1 ...
  1 | 1 2 5  15    52       203           877             4140 ...
  2 | 1 2 8  59   652      9736        186478          4421018 ...
  3 | 1 2 8 109  3623    200522      16514461       1912959395 ...
  4 | 1 2 8 128 11087   2232875     775098224     428188962261 ...
  5 | 1 2 8 128 21380  15312665   22165394234   57353442460140 ...
  6 | 1 2 8 128 29228  70197998  422059040480 5051078354829005 ...
  7 | 1 2 8 128 32297 227731312 5686426671375 ...
      ...
The T(1,2) = 5 set systems are:
  {},
  {{1,2}},
  {{1,2}, {2}},
  {{1},{1,2}},
  {{1}, {2}}.

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

Rows n=0..4 are A000012, A000110, A178165, A178171, A178173.
Cf. A058891, A188445, A219585, A219727.
Main diagonal gives A374573.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); (vecsum(WeighT(v)) + 1)^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)}

Lim_{n->oo} A(n,k) = 2^k.

A322487 Number of (3*n) X n matrices with nonnegative integer entries and each column sum being 3 up to permutation of rows.

Original entry on oeis.org

1, 3, 31, 686, 27036, 1688360, 154703688, 19692332568, 3342458334775, 732812082630803, 202322386045180686, 68898094282978653925, 28443422251718020038049, 14029033632468285836567998, 8164217197799501761637725983, 5545466507405459243366712102466

Offset: 0

Andrew Howroyd, Dec 11 2018

nonn

Also number of multiset partitions of [1,1,1,2,2,2,...,n,n,n] into nonempty multisets. - Marko Riedel, Nov 29 2022

			a(1) = 3 because up to permutations of rows there are 3 column vectors with sum 3: [1, 1, 1], [2, 1, 0] and [3, 0, 0].

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

Row n=3 of A219727.

Cf. A165434, A358721.

A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120

Offset: 0

Gus Wiseman, Dec 26 2018

nonn

A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020

			The a(1) = 1 through a(4) = 29 multiset partitions:
  {{1}}   {{1,1}}     {{1,1,1}}       {{1,1,1,1}}
          {{1,2}}     {{1,2,3}}       {{1,1,2,2}}
         {{1},{1}}   {{1},{1,1}}      {{1,2,3,4}}
         {{1},{2}}   {{1},{2,3}}     {{1},{1,1,1}}
                     {{2},{1,3}}     {{1,1},{1,1}}
                     {{3},{1,2}}     {{1},{1,2,2}}
                    {{1},{1},{1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}    {{1,2},{1,2}}
                                     {{1},{2,3,4}}
                                     {{1,2},{3,4}}
                                     {{1,3},{2,4}}
                                     {{1,4},{2,3}}
                                     {{2},{1,1,2}}
                                     {{2},{1,3,4}}
                                     {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                    {{1},{1},{1,1}}
                                    {{1},{1},{2,2}}
                                    {{1},{2},{1,2}}
                                    {{1},{2},{3,4}}
                                    {{1},{3},{2,4}}
                                    {{1},{4},{2,3}}
                                    {{2},{2},{1,1}}
                                    {{2},{3},{1,4}}
                                    {{2},{4},{1,3}}
                                    {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A322786.

Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.

u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Sum[u[Array[Prime,d,1,Times]^(n/d),Array[Prime,d,1,Times]^(n/d)],{d,Divisors[n]}],{n,12}]

a(n) = Sum_{d|n} A001055(A002110(n/d)^d).

a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020

a(14)-a(15) from Alois P. Heinz, Jan 16 2019

Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020

A256384 Number A(n,k) of factorizations of m^n into at most n factors, where m is a product of exactly k distinct primes; 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, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 19, 5, 1, 1, 1, 41, 171, 74, 7, 1, 1, 1, 122, 1675, 1975, 248, 11, 1, 1, 1, 365, 16683, 64182, 20096, 814, 15, 1, 1, 1, 1094, 166699, 2203215, 2213016, 187921, 2457, 22, 1, 1, 1, 3281, 1666731, 76727374, 268446852, 69406700, 1609727, 7168, 30, 1

Offset: 0

Alois P. Heinz, Mar 27 2015

A(n,k) is also the number of k-partite partitions of (n)^k into at most n k-tuples. A(2,2) = 5: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(1,2),(1,0)], [(1,1),(1,1)].

			A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into at most 2 factors: 36, 2*18, 3*12, 4*9, 6*6.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,         1, ...
  1, 1,   1,     1,       1,         1, ...
  1, 2,   5,    14,      41,       122, ...
  1, 3,  19,   171,    1675,     16683, ...
  1, 5,  74,  1975,   64182,   2203215, ...
  1, 7, 248, 20096, 2213016, 268446852, ...

Columns k=0-3 give: A000012, A000041, A254686, A254811.
Rows n=0+1,2-3 give: A000012, A007051, A256493.
Cf. A219727.

b[n_, k_, i_] := b[n, k, i] = If[n>k, 0, 1] + If[PrimeQ[n] || i<2, 0, Sum[ If[d > k, 0, b[n/d, d, i-1]], {d, Divisors[n][[2 ;; -2]]}]]; A[0, ] = 1; A[1, ] = 1; A[, 0] = 1; A[n, k_] := With[{t = Times @@ Prime[ Range[k] ]}, b[t^n, t^n, n]]; Table[diag = Table[A[n-k, k], {k, n, 0, -1}]; Print[ diag]; diag, {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A219678 Number of tripartite partitions of (n,n,n) into triples.

Original entry on oeis.org

1, 5, 66, 686, 6721, 58616, 476781, 3608348, 25781989, 174810494, 1132328205, 7037425172, 42140788751, 243918472743, 1368647208107, 7462686474948, 39626100989332, 205283558905562, 1039263967957447, 5149048724566723, 24998922141116056, 119073277412589351

Offset: 0

Alois P. Heinz, Nov 26 2012

nonn

a(n) is also the number of factorizations of m^n where m is a product of 3 distinct primes. a(1) = 5: (2*3*5)^1 = 30 has 5 factorizations: 30, 5*6, 3*10, 2*3*5, 2*15.

			a(1) = 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)].

Sean A. Irvine, Table of n, a(n) for n = 0..29

Column k=3 of A219727.

Cf. A219560.

More terms from Sean A. Irvine, Aug 14 2014

A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 5, 3, 15, 9, 5, 52, 7, 203, 66, 31, 11, 877, 15, 4140, 712, 109, 22, 21147, 686, 30, 115975, 10457, 339, 42, 678570, 56, 4213597, 198091, 27036, 6721, 1043, 77, 27644437, 101, 190899322, 4659138, 2998, 135, 1382958545, 1688360, 58616, 176

Offset: 1

Gus Wiseman, Dec 26 2018

			Triangle begins:
        1
        2       2
        5       3
       15       9       5
       52       7
      203      66      31      11
      877      15
     4140     712     109      22
    21147     686      30
   115975   10457     339      42
   678570      56
  4213597  198091   27036    6721    1043      77
For example, row 4 counts the following multiset partitions.
  {{1,2,3,4}}        {{1,1,2,2}}        {{1,1,1,1}}
  {{1},{2,3,4}}      {{1},{1,2,2}}      {{1},{1,1,1}}
  {{1,2},{3,4}}      {{1,1},{2,2}}      {{1,1},{1,1}}
  {{1,3},{2,4}}      {{1,2},{1,2}}      {{1},{1},{1,1}}
  {{1,4},{2,3}}      {{2},{1,1,2}}      {{1},{1},{1},{1}}
  {{2},{1,3,4}}      {{1},{1},{2,2}}
  {{3},{1,2,4}}      {{1},{2},{1,2}}
  {{4},{1,2,3}}      {{2},{2},{1,1}}
  {{1},{2},{3,4}}    {{1},{1},{2},{2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (first 50 rows)

Row sums are A322784. First column is A000110.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.

u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Table[u[Array[Prime,n/d,1,Times]^d,Array[Prime,n/d,1,Times]^d],{d,Divisors[n]}],{n,10}]

\\ needs T(n,k) from A219727.
Row(n)={[T(d,n/d) | d<-divisors(n)]}
{ for(n=1, 12, print(Row(n))) } \\ Andrew Howroyd, Jan 11 2020

T(n,k) = A001055(A002110(n/d)^d), where d = A027750(n,k).

T(n,k) = A219727(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Jan 11 2020

Edited by Peter Munn, Mar 05 2025

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.

A322488 Number of factorizations of the n-th power of the product of the first n primes.

Original entry on oeis.org

1, 1, 9, 686, 911838, 27119992809, 23970534519938280, 790361842548583118561351, 1186709456459520739315771458325336, 96786580459441954551347685958203256606168610, 502265575410823018475962732653887865889973989497971475955

Offset: 0

Alois P. Heinz, Dec 11 2018

nonn

Also number of n-partite partitions of {n}^n into n-tuples. a(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)].

Also number of partitions of the multiset with n copies each of 1,2,...,n. a(2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.

			a(2) = 9: (2*3)^2 = A002110(2)^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.

Main diagonal of A219727.

Cf. A000040, A000110, A001055, A002110, A181555.

a(n) = A001055(A181555(n)) = A001055(A002110(n)^n).

a(10) from Andrew Howroyd, Dec 11 2018

cp's OEIS Frontend

A330964 Array read by antidiagonals: A(n,k) is the number of sets of nonempty subsets of a k-element set where each element appears in at most n subsets.

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A322487 Number of (3*n) X n matrices with nonnegative integer entries and each column sum being 3 up to permutation of rows.

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A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

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A256384 Number A(n,k) of factorizations of m^n into at most n factors, where m is a product of exactly k distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A219678 Number of tripartite partitions of (n,n,n) into triples.

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A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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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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A322488 Number of factorizations of the n-th power of the product of the first n primes.

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