A020555 -id:A020555 - cp's OEIS frontend

A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

1, 1, 2, 6, 15, 40, 121

Offset: 0

Gus Wiseman, Jul 18 2018

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows and no equal columns.

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 6 strict multiset partitions with no equivalent vertices (first column) and their duals (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(2)(3) <-> (1)(2)(3)

Cf. A000009, A001055, A007716, A007717, A020555, A045778.

Cf. A316974, A316978, A316979, A316980, A316983.

A322764 Number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_n, and 2 copies each of y_1 and y_2.

Original entry on oeis.org

9, 26, 92, 371, 1663, 8155, 43263, 246218, 1493344, 9600683, 65133513, 464538351, 3471671717, 27109690422, 220646396816, 1867649896679, 16408260807503, 149357276866099, 1406334890073883, 13677748330883790, 137221985081833892

Offset: 0

N. J. A. Sloane, Dec 30 2018

nonn

The initial 9 is also A020555(2).

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000110 (Bell number), A020555, A322773.

Column 2 of the array in A322765.

T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2);
vector(20, n, T(n-1, 2)) \\ Seiichi Manyama, Nov 21 2020

4*a(n) = 3*b(n) + 2*b(n+1) + 3*b(n+2) + 2*b(n+3) + b(n+4), where b(n) = A000110(n). - Seiichi Manyama, Nov 21 2020

A358710 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 2, 2, ..., n, n] into k nonempty submultisets, for 1 <= k <= 2n.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 1, 13, 26, 19, 6, 1, 1, 40, 183, 259, 163, 55, 10, 1, 1, 121, 1190, 3115, 3373, 1896, 620, 125, 15, 1, 1, 364, 7443, 34891, 62240, 54774, 27610, 8706, 1795, 245, 21, 1, 1, 1093, 45626, 374059, 1072316, 1435175, 1063570, 485850, 146363, 30261, 4361, 434, 28, 1

Offset: 0

Marko Riedel, Nov 27 2022

A generalization of ordinary Stirling set numbers to multisets that contain some m instances each of n elements, here we have m=2.

			The triangular array starts:
[0] 1;
[1] 1,   1;
[2] 1,   4,    3,     1;
[3] 1,  13,   26,    19,     6,     1;
[4] 1,  40,  183,   259,   163,    55,    10,    1;
[5] 1, 121, 1190,  3115,  3373,  1896,   620,  125,   15,   1;
[6] 1, 364, 7443, 34891, 62240, 54774, 27610, 8706, 1795, 245, 21, 1;

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

Sidney Cadot, Table of n, a(n) for n = 0..420 (terms 1..420 from Marko Riedel)
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, A020555 (row sums), A358721, A358722.

A316892 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices.

Original entry on oeis.org

1, 1, 3, 9, 24, 69, 211, 654

Offset: 0

Gus Wiseman, Jul 18 2018

Also the number of unlabeled graphs with n edges, allowing loops, with no equivalent vertices (two vertices are equivalent if in every edge the multiplicity of the first is equal to the multiplicity of the second). For example, non-isomorphic representatives of the a(2) = 3 multigraphs are {(1,2),(1,3)}, {(1,1),(1,2)}, {(1,1),(2,2)}.

			Non-isomorphic representatives of the a(3) = 9 strict multiset partitions:
  (112)(233)
  (1)(2)(1233)
  (1)(12)(233)
  (2)(11)(233)
  (11)(22)(33)
  (12)(13)(23)
  (1)(2)(3)(123)
  (1)(2)(12)(33)
  (1)(2)(13)(23)

Cf. A007716, A007717, A020555, A050535, A053419, A094574, A316974.

a(6)-a(7) from Andrew Howroyd, Feb 07 2020

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.

A020559 Number of ordered multigraphs on n labeled edges (with loops).

Original entry on oeis.org

1, 2, 11, 97, 1219, 20385, 433022, 11296844, 352866598, 12938878499, 548257129281, 26503637228615, 1446212232918009, 88278080019931590, 5981590442549971867, 446907535344317788261, 36602523445840041088223

Offset: 0

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A020555, A020558.

E.g.f.: exp((3*x-2)/(2-2*x))*Sum_{n>=0}1/(n!*(1-x)^binomial(n+1, 2)). - Vladeta Jovovic, May 02 2004

a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n, k) * A020555(k). - Sean A. Irvine, Apr 24 2019

A316977 Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 12, 575, 66080, 13830706, 4566898564, 2181901435364, 1422774451251512, 1213875872220833664, 1312273759143855989808, 1752860078230602866012288, 2834766624822130489716563008, 5458358420687156358967526721408, 12339106957086349462329140342122112

Offset: 1

Gus Wiseman, Jul 17 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(2) = 12 trees are (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122).

Cf. A000081, A000669, A007717, A020555, A050535, A094574, A292504, A316655, A316972, A316974.

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]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[gro[Ceiling[Range[1/2,n,1/2]]]],{n,4}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(2*n), vars=vector(2*n-2,i,sv(2+i))); v[1]=sv(1); for(n=2, #v, v[n] = substvec(polcoef( sExp(x*Ser(v[1..n])), n ), vars[1..n-2], vector(n-2))); sCartProd(x*Ser(v), 1/(1-x^2*symGroupCycleIndex(2)) + O(x*x^(2*n)))}
seq(n)={my(p=substvec(cycleIndexSeries(n), [sv(1), sv(2)], [1,1])); vector(n, n, polcoef(p,2*n))} \\ Andrew Howroyd, Jan 02 2021

a(n) = A292505(A061742(n)). - Andrew Howroyd, Nov 19 2018

Terms a(6) and beyond from Andrew Howroyd, Jan 02 2021

cp's OEIS Frontend

A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

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A322764 Number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_n, and 2 copies each of y_1 and y_2.

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A358710 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 2, 2, ..., n, n] into k nonempty submultisets, for 1 <= k <= 2n.

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A316892 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices.

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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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A020559 Number of ordered multigraphs on n labeled edges (with loops).

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A316977 Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.

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