A255906 -id:A255906 - cp's OEIS frontend

A317075 Number of connected antichains of multisets with multiset-join a normal multiset of size n.

1, 1, 2, 10, 147, 8998

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is normal if it spans an initial interval of positive integers. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			The a(3) = 10 connected antichains of multisets:
  (111),
  (122), (12)(22),
  (112), (11)(12),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A048143, A007718, A255906, A286520, A303837, A303838, A304716, A305001, A305078.

Cf. A317073, A317076, A317077, A317080.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
Table[Length[Join@@Table[cuu[m],{m,allnorm[n]}]],{n,5}]

A317076 Number of connected antichains of multisets with multiset-join a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 8, 110, 7047

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			The a(3) = 8 connected antichains of multisets:
  (111),
  (112), (11)(12),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A048143, A255906, A286520, A303837, A303838, A304716, A305001, A305078.

Cf. A317074, A317075, A317078, A317080.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
Table[Length[Join@@Table[cuu[m],{m,strnorm[n]}]],{n,5}]

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 8, 7, 7, 8, 11, 12, 15, 5, 15, 17, 22, 14, 27, 19, 30, 13, 27, 30, 33, 26, 42, 37, 56, 7, 44, 45, 51, 34, 77, 67, 72, 25

Offset: 1

Gus Wiseman, Aug 28 2018

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

The prime indices of n are the n-th row of A296150.

			The a(18) = 17 combinatory separations of {1,1,2,2,3}:
  {11223}
  {1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
  {1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
  {1,1,1,11} {1,1,1,12}
  {1,1,1,1,1}

Cf. A007716, A056239, A255906, A269134, A296150, A305936.

Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize,mps[nrmptn[n]],{2}]]],{n,20}]

A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

Original entry on oeis.org

1, 4, 10, 32, 80, 239, 605, 1670, 4251

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318563, A318564, A318565, A318566.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
strnormQ[m_]:=OrderedQ[Length/@Split[m],GreaterEqual];
Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@strnormQ/@#[[2]]&]],{n,6}]

A320463 MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 113, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 3277, 4537, 4859, 5249, 5311, 7423, 8249, 8507, 16211, 16403, 16559, 16783, 16837, 17719, 20443, 20453, 24553, 25477, 26273, 26969, 27521, 34567, 37439, 39437, 41689, 42011, 42137, 42601, 43873, 43957

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    113: {{1,2,3}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1291: {{1,2,3,4}}
   1363: {{1,3},{2,3}}
   1469: {{1,2},{1,2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
   3277: {{1,3},{1,2,3}}
   4537: {{1,2},{1,3,4}}
   4859: {{1,4},{1,2,3}}
   5249: {{1,3},{1,2,4}}
   5311: {{2,3},{1,2,3}}
   7423: {{1,2},{2,3,4}}
   8249: {{2,4},{1,2,3}}
   8507: {{2,3},{1,2,4}}
  16211: {{1,2},{1,3},{1,4}}

Wikipedia, Hypergraph

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320458, A320464, A320532, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]

A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 113, 169, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 2197, 3277, 4537, 4859, 4901, 5249, 5311, 7423, 7943, 8249, 8507, 10933, 12769, 16211, 16403, 16559, 16783, 16837, 17719, 19097, 20443, 20453, 24553, 25181, 25477, 26273, 26969, 27521, 28561, 28717

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
     1: {}
    13: {{1,2}}
   113: {{1,2,3}}
   169: {{1,2},{1,2}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
  1291: {{1,2,3,4}}
  1363: {{1,3},{2,3}}
  1469: {{1,2},{1,2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2117: {{1,3},{2,4}}
  2197: {{1,2},{1,2},{1,2}}
  3277: {{1,3},{1,2,3}}
  4537: {{1,2},{1,3,4}}
  4859: {{1,4},{1,2,3}}
  4901: {{1,2},{1,2},{1,3}}
  5249: {{1,3},{1,2,4}}
  5311: {{2,3},{1,2,3}}
  7423: {{1,2},{2,3,4}}
  7943: {{1,2},{1,2},{2,3}}
  8249: {{2,4},{1,2,3}}
  8507: {{2,3},{1,2,4}}

Wikipedia, Hypergraph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320459, A320463, A320532, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]

A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

Original entry on oeis.org

1, 1, 2, 12, 112, 1444, 24099, 492434, 11913985

Offset: 0

Gus Wiseman, Dec 26 2019

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).
A finite multiset is normal if it covers an initial interval of positive integers.
First differs from A316651 at a(6) = 24099, A316651(6) = 24086. For example, ((1(12))(2(11))) and ((2(11))(1(12))) are considered identical for A316651 (series-reduced rooted trees), but {{{1},{1,2}},{{2},{1,1}}} and {{{2},{1,1}},{{1},{1,2}}} are different series/singleton-reduced rooted trees.

			The a(0) = 1 through a(3) = 12 trees:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

The orderless version is A316651.
The strongly normal case is A330471.
The unlabeled version is A330470.
The balanced version is A330655.
The case with all atoms distinct is A000311.
The case with all atoms equal is A196545.
Normal multiset partitions are A255906.
Cf. A000669, A004114, A005804, A281118, A316651, A330469, A330626, A330676.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[ssrtrees[s]],{s,allnorm[n]}],{n,0,5}]

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093

Offset: 1

Andrew Howroyd, Jan 23 2020

nonn

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
From Gus Wiseman, Apr 03 2025: (Start)
Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{1,2}}
         {{1},{2}}  {{1},{1},{2}}  {{1,2},{1,3}}
                    {{1},{2},{2}}  {{1,2},{2,3}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{1,3},{2,4}}
                                   {{1,4},{2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{2},{3}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A330942, A331639.
For constant instead of strict blocks we have A034729.
Without equal sizes we have A116540 (normal set multipartitions).
Without strict blocks we have A317583.
For distinct instead of equal sizes we have A382428, non-strict blocks A326517.
For equal sums instead of sizes we have A382429, non-strict blocks A326518.
Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.
Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.

a(n) = Sum_{d|n} A330942(n/d, d).

a(p) = 2^(p-1) + 1 for prime p.

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

For prime indices of prime indices we have A320456.
For binary indices of binary indices we have A326754.
An opposite version is A371292, A371293.
The case with squarefree product of prime indices is A371448.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],normQ[Join@@bpe/@prix[#]]&]

A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9

Offset: 0

Gus Wiseman, Mar 30 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}

For constant instead of strict blocks we have A000045.
Factorizations of this type are counted by A050326, with distinct sums A381633.
For the strong case see A292444, A382430, complement A381996, A382523.
MM-numbers of sets of sets are A302494, see A302478, A382201.
Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.
For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).
With distinct sums we have A382459.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n], Length[Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]

cp's OEIS Frontend

A317075 Number of connected antichains of multisets with multiset-join a normal multiset of size n.

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A317076 Number of connected antichains of multisets with multiset-join a strongly normal multiset of size n.

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A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

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A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

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A320463 MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.

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A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.

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A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

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A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

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A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

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