A049311 -id:A049311 - cp's OEIS frontend

A319752 Number of non-isomorphic intersecting multiset partitions of weight n.

1, 1, 3, 6, 16, 35, 94, 222, 584, 1488, 3977

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 16 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{2},{1,2,2}}
  {{3},{1,2,3}}
  {{1,1},{1,1}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,3},{2,3}}
  {{1},{1},{1,1}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A319616.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319789.

A101370 Number of zero-one matrices with n ones and no zero rows or columns.

Original entry on oeis.org

1, 4, 24, 196, 2016, 24976, 361792, 5997872, 111969552, 2324081728, 53089540992, 1323476327488, 35752797376128, 1040367629940352, 32441861122796672, 1079239231677587264, 38151510015777089280, 1428149538870997774080, 56435732691153773665280

Offset: 1

Peter J. Cameron, Jan 14 2005

a(n) = (1/(4*n!)) * Sum_{r, s>=0} (r*s)_n / 2^(r+s), where (m)_n is the falling factorial m * (m-1) * ... * (m-n+1). [Maia and Mendez]

			a(2)=4:
[1 1] [1] [1 0] [0 1]
..... [1] [0 1] [1 0]
From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 24 matrices:
  [111]
.
  [11][11][110][101][10][100][011][01][010][001]
  [10][01][001][010][11][011][100][11][101][110]
.
  [1][10][10][10][100][100][01][01][010][01][010][001][001]
  [1][10][01][01][010][001][10][10][100][01][001][100][010]
  [1][01][10][01][001][010][10][01][001][10][100][010][100]
(End)

Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, p. 435 (IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13), Springer, Berlin. [Rainer Rosenthal, Apr 10 2007]

Alois P. Heinz, Table of n, a(n) for n = 1..400
P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
Giulio Cerbai and Anders Claesson, Enumerative aspects of Caylerian polynomials, arXiv:2411.08426 [math.CO], 2024. See pp. 3, 19.
Loïc Foissy, Claudia Malvenuto, and Frédéric Patras, Matrix symmetric and quasi-symmetric functions and noncommutative representation theory, arXiv:2503.14417 [math.CO], 2025. See p. 20.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Cf. A000670 (the sequence P(n)), A049311 (row and column permutations allowed), A120733, A122725, A135589, A283877, A321446, A321587.

P:=function(n) return Sum([1..n],x->Stirling2(n,x)*Factorial(x)); end;

F:=function(n) return Sum([1..n],x->(-1)^(n-x)*Stirling1(n,x)*P(x)^2)/Factorial(n); end;

m = 17; a670[n_] = Sum[ StirlingS2[n, k]*k!, {k, 0, n}]; Rest[ CoefficientList[ Series[ Sum[ a670[n]^2*(Log[1 + x]^n/n!), {n, 0, m}], {x, 0, m}], x]] (* Jean-François Alcover, Sep 02 2011, after g.f.  *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#]]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

{A000670(n)=sum(k=0,n,stirling(n, k,2)*k!)}
{a(n)=polcoeff(sum(m=0,n,A000670(m)^2*log(1+x+x*O(x^n))^m/m!),n)}
/* Paul D. Hanna, Nov 07 2009 */

a(n) = (Sum s(n, k) * P(k)^2)/n!, where P(n) is the number of labeled total preorders on {1, ..., n} (A000670), s are signed Stirling numbers of the first kind.
G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^m. - Vladeta Jovovic, Mar 25 2006
Inverse binomial transform of A007322. - Vladeta Jovovic, Aug 17 2006
G.f.: Sum_{n>=0} 1/(2-(1+x)^n)/2^(n+1). - Vladeta Jovovic, Sep 23 2006
G.f.: Sum_{n>=0} A000670(n)^2*log(1+x)^n/n! where 1/(1-x) = Sum_{n>=0} A000670(n)*log(1+x)^n/n!. - Paul D. Hanna, Nov 07 2009
a(n) ~ n! / (2^(2+log(2)/2) * (log(2))^(2*(n+1))). - Vaclav Kotesovec, Dec 31 2013

A303546 Number of non-isomorphic aperiodic multiset partitions of weight n.

Original entry on oeis.org

1, 3, 9, 29, 90, 285, 909, 2984, 9935, 34113, 119368, 428923, 1574223, 5915235, 22699730, 89000042, 356058539, 1453069854, 6044132793, 25612564200, 110503626702, 485161228675, 2166488899641, 9835209480533, 45370059225227

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime. For this sequence neither the parts nor their multiset union are required to be aperiodic, only the multiset of parts.

			Non-isomorphic representatives of the a(3) = 9 aperiodic multiset partitions are:
  {{1,1,1}}, {{1,2,2}}, {{1,2,3}},
  {{1},{1,1}}, {{1},{2,2}}, {{1},{2,3}}, {{2},{1,2}},
  {{1},{2},{2}}, {{1},{2},{3}}.

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

Cf. A000740, A000837, A001055, A007716, A007916, A049311, A100953, A255906, A269134, A275024, A293993, A301700, A303386, A303431, A303547.

a(n) = Sum_{d|n} mu(d) * A007716(n/d).

A306006 Number of non-isomorphic intersecting set-systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 16, 30, 57, 109, 209, 431, 873, 1850, 3979, 8819, 19863

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of S is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 10 set-systems:
{{1,2,3,4,5,6}}
{{5},{1,2,3,4,5}}
{{1,5},{2,3,4,5}}
{{3,4},{1,2,3,4}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{3},{2,3},{1,2,3}}
{{4},{1,4},{2,3,4}}
{{1,2},{1,3},{2,3}}
{{1,4},{2,4},{3,4}}

Cf. A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

a(10)-a(17) from Bert Dobbelaere, May 04 2025

A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

Original entry on oeis.org

1, 1, 3, 29, 1885, 18658259

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated elements).

			Non-isomorphic representatives of the a(3) = 29 covers:
   {{1,3},{2,3}}
   {{1},{2},{3}}
   {{1},{3},{2,3}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{3},{1,3},{2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2,3}}
   {{1},{2},{1,3},{2,3}}
   {{2},{3},{1,3},{2,3}}
   {{1},{3},{2,3},{1,2,3}}
   {{2},{3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3}}
   {{2},{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,3},{2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{3},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A316980, A316983, A318099, A319558, A319559, A319616-A319646, A300913.

a(5) from Max Alekseyev, Jul 13 2022

A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 7, 8, 15, 18, 30, 37, 58, 71, 105, 131, 186, 230, 318, 393, 530, 653, 863, 1060, 1380, 1686, 2164, 2637, 3345, 4057, 5096, 6158, 7665, 9228, 11395, 13671, 16765, 20040, 24418, 29098, 35251, 41869, 50460, 59755, 71669, 84626, 101050

Offset: 0

Henry Bottomley, Aug 01 2005

nonn

Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018

			a(5) = 3 since 5 can be partitioned as 1+1+1+1+1, 2+1+1+1, or 2+2+1; not counted are 5, 4+1, or 3+2.
a(6) = 7 since 6 can be partitioned as 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 1+2+3, 3+3; not counted are 1+1+4, 2+4, 1+5, 6.
From _Gus Wiseman_, Oct 30 2018: (Start)
The a(2) = 1 through a(8) = 15 partitions with no part larger than n/2:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (2111)   (222)     (331)      (332)
               (1111)  (11111)  (321)     (2221)     (422)
                                (2211)    (3211)     (431)
                                (3111)    (22111)    (2222)
                                (21111)   (31111)    (3221)
                                (111111)  (211111)   (3311)
                                          (1111111)  (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(2) = 1 through a(8) = 15 partitions into n/2 or fewer parts:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (22)  (32)  (33)   (43)   (44)
            (31)  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (222)  (322)  (71)
                        (321)  (331)  (332)
                        (411)  (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (2222)
                                      (3221)
                                      (3311)
                                      (4211)
                                      (5111)
The a(6) = 7 integer partitions of 6 with no part larger than n/2 together with a realizing set multipartition of each (the parts of the partition count the appearances of each vertex in the set multipartition):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2},{1,2,3,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}
(End)

Cf. A000070, A000569, A025065, A049311, A096373, A116540, A147878, A209816, A283877, A306005, A320921.

A000070 := proc(n) add( combinat[numbpart](i),i=0..n) ; end proc:
A110618 := proc(n) combinat[numbpart](n) - A000070(floor((n-1)/2)) ; end proc: # R. J. Mathar, Jan 24 2011

f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; g[n_] := f[n, Floor[n/2]]; g[0] = 1; g[1] = 0; Array[g, 47, 0] (* Robert G. Wilson v, Jan 23 2011 *)
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]]]];
multhyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multhyp[#]!={}&]],{n,8}] (* Gus Wiseman, Oct 30 2018 *)

a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018

a(n) = A000041(n) - Sum_{i=0..floor((n-1)/2)} A000041(i) = A000041(n) - A000070(floor((n-1)/2)) = A110619(n, 2).

a(2*n) = A209816(n). - Gus Wiseman, Oct 30 2018

A321405 Number of non-isomorphic self-dual set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 9, 16, 28, 47

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of (0,1) symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 16 set systems:
  {{1}}  {{1}{2}}  {{2}{12}}    {{1}{3}{23}}    {{2}{13}{23}}
                   {{1}{2}{3}}  {{1}{2}{3}{4}}  {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}
.
  {{12}{13}{23}}        {{13}{23}{123}}          {{1}{13}{14}{234}}
  {{3}{23}{123}}        {{1}{23}{24}{34}}        {{12}{13}{24}{34}}
  {{1}{3}{24}{34}}      {{1}{4}{34}{234}}        {{1}{24}{34}{234}}
  {{2}{4}{12}{34}}      {{2}{13}{24}{34}}        {{2}{14}{34}{234}}
  {{1}{2}{3}{5}{45}}    {{3}{4}{14}{234}}        {{3}{4}{134}{234}}
  {{1}{2}{3}{4}{5}{6}}  {{1}{2}{4}{35}{45}}      {{4}{13}{14}{234}}
                        {{1}{3}{5}{23}{45}}      {{1}{2}{34}{35}{45}}
                        {{1}{2}{3}{4}{6}{56}}    {{1}{2}{5}{45}{345}}
                        {{1}{2}{3}{4}{5}{6}{7}}  {{1}{3}{24}{35}{45}}
                                                 {{1}{4}{5}{25}{345}}
                                                 {{2}{4}{12}{35}{45}}
                                                 {{4}{5}{13}{23}{45}}
                                                 {{1}{2}{3}{5}{46}{56}}
                                                 {{1}{2}{4}{6}{34}{56}}
                                                 {{1}{2}{3}{4}{5}{7}{67}}
                                                 {{1}{2}{3}{4}{5}{6}{7}{8}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A316980, A316983, A319616.

Cf. A320796, A320797, A321401, A321403, A321406.

A292432 Number of normal multisets that cannot be expressed as the multiset-union of a set of sets.

Original entry on oeis.org

0, 1, 1, 3, 5, 9, 16, 27, 46, 76, 130, 203, 350, 554, 890, 1474, 2285, 3732, 5852, 9297, 14628, 22936, 35903, 55893, 86967, 134585, 207934, 321122, 492634, 757490

Offset: 1

Gus Wiseman, Oct 02 2017

A multiset is normal if it spans an initial interval of positive integers. Most normal multisets can be expressed as the multiset-union of a set of sets. For example, {1,1,2,2} is the multiset-union of {{1},{2},{1,2}}.

			The a(6) = 9 multisets are: {1,1,1,1,1,1}, {1,1,1,1,1,2}, {1,1,1,1,2,2}, {1,1,1,1,2,3}, {1,1,1,2,2,2}, {1,1,2,2,2,2}, {1,2,2,2,2,2}, {1,2,2,2,2,3}, {1,2,3,3,3,3}.

Cf. A049311, A089259, A116540, A283877, A292444.

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025

A292444 Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 30, 39, 50, 65, 83, 105, 131, 167, 207, 257, 317, 391, 478, 585, 708, 864, 1037, 1252, 1498

Offset: 1

Gus Wiseman, Oct 02 2017

Non-isomorphic finite multisets correspond to integer partitions. For example, the partition (3221) corresponds to the multiset {1,1,1,2,2,3,3,4}.

			Representatives of the a(7) = 6 multisets are: {1,1,1,1,1,1,1}, {1,1,1,1,1,1,2}, {1,1,1,1,1,2,2}, {1,1,1,1,1,2,3}, {1,1,1,1,2,2,2}, {1,1,1,1,2,2,3}.

Cf. A049311, A089259, A116540, A283877, A292432.

a(12)-a(30) from Bert Dobbelaere, Mar 30 2025

A104602 Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns.

Original entry on oeis.org

1, 1, 2, 10, 70, 642, 7246, 97052, 1503700, 26448872, 520556146, 11333475922, 270422904986, 7016943483450, 196717253145470, 5925211960335162, 190825629733950454, 6543503207678564364, 238019066600097607402, 9153956822981328930170, 371126108428565106918404

Offset: 0

Ralf Stephan, Mar 27 2005

nonn

Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns, up to row and column permutation, is A057151(n). - Vladeta Jovovic, Mar 25 2006

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 10 matrices:
  [1 1] [1 1] [1 0] [0 1]
  [1 0] [0 1] [1 1] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Row sums of triangle A104601.

Cf. A048291, A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

Table[1/n!*Sum[StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}],{k,0,n}],{n,1,20}] (* Vaclav Kotesovec, May 07 2014 *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A048144(k). - Vladeta Jovovic, Mar 25 2006
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.28889864564457451375789435201798... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 1 / (log(2) * 2^(log(2)/2+2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n*(n+1)). - Paul D. Hanna, Mar 26 2018

More terms from Vladeta Jovovic, Mar 25 2006

a(0)=1 prepended by Alois P. Heinz, Jan 14 2015

cp's OEIS Frontend

A319752 Number of non-isomorphic intersecting multiset partitions of weight n.

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A101370 Number of zero-one matrices with n ones and no zero rows or columns.

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A303546 Number of non-isomorphic aperiodic multiset partitions of weight n.

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A306006 Number of non-isomorphic intersecting set-systems of weight n.

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A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

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A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

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A321405 Number of non-isomorphic self-dual set systems of weight n.

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A292432 Number of normal multisets that cannot be expressed as the multiset-union of a set of sets.

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A292444 Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.

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