A317791 -id:A317791 - cp's OEIS frontend

A323794 Number of non-isomorphic weight-n multisets of sets of multisets.

1, 1, 5, 17, 77, 318, 1561, 7667

Offset: 0

Gus Wiseman, Jan 28 2019

Also the number of non-isomorphic set multipartitions of multiset partitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 17 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{1}}  {{1}{11}}
         {{1}}{{2}}  {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Cf. A007716, A049311, A283877, A306186, A316980, A317791, A318565, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323795.

A318287 Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 3, 4, 5, 3, 7, 4, 7, 9, 5, 5, 12, 6, 12, 14, 10, 8, 13, 12, 14, 14, 18, 10, 34

Offset: 1

Gus Wiseman, Aug 23 2018

			Non-isomorphic representatives of the a(20) = 12 strict multiset partitions of {1,1,1,2,3}:
  {{1,1,1,2,3}}
  {{1},{1,1,2,3}}
  {{2},{1,1,1,3}}
  {{1,1},{1,2,3}}
  {{1,2},{1,1,3}}
  {{2,3},{1,1,1}}
  {{1},{2},{1,1,3}}
  {{1},{1,1},{2,3}}
  {{1},{1,2},{1,3}}
  {{2},{3},{1,1,1}}
  {{2},{1,1},{1,3}}
  {{1},{2},{3},{1,1}}

Cf. A001055, A007716, A045778, A181821, A305936, A316980, A317533, A317776, A317791.

Cf. A318283, A318284, A318285, A318286, A318357, A318362, A318371.

a(n) = A318357(A181821(n)).

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 5, 1, 2, 3, 5, 1, 7, 1, 5, 3, 2, 1, 9, 4, 2, 8, 5, 1, 10

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(12) = 5 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{1},{2},{3}}

Cf. A007716, A049311, A116540, A181821, A317791.

Cf. A318283, A318285, A318287, A318360, A318361, A318369, A318371.

a(n) = A318369(A181821(n)).

A318369 Number of non-isomorphic set multipartitions (multisets of sets) of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(180) = 7 set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{3},{1,2},{1,2}}
  {{1},{1},{2},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{1},{2},{2},{3}}

Cf. A001055, A007716, A056239, A112798, A049311, A116540, A317791, A318360, A318361.

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}]

A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 2, 10, 11, 20, 15, 90, 51, 80, 6, 468, 93, 2910, 80, 521, 277, 20644, 80, 334, 1761, 393, 521, 165874, 1374

Offset: 1

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Non-isomorphic representatives of the a(2) = 1 through a(9) = 10 multisystems (commas and outer brackets elided):
    1  11  12  111      112      1111            123      1122
               {1}{11}  {1}{12}  {1}{111}        {1}{23}  {1}{122}
                        {2}{11}  {11}{11}                 {11}{22}
                                 {1}{1}{11}               {12}{12}
                                 {{1}}{{1}{11}}           {1}{1}{22}
                                 {{11}}{{1}{1}}           {1}{2}{12}
                                                          {{1}}{{1}{22}}
                                                          {{11}}{{2}{2}}
                                                          {{1}}{{2}{12}}
                                                          {{12}}{{1}{2}}
Non-isomorphic representatives of the a(12) = 15 multisystems:
  {1,1,2,3}
  {{1},{1,2,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{2},{1,1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{{1}},{{1},{2,3}}}
  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,3}}}
  {{{1,2}},{{1},{3}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}
  {{{2,3}},{{1},{1}}}

The labeled version is A318846.
The maximum-depth version is A330664.
Unlabeled balanced reduced multisystems by weight are A330474.
The case of constant or strict atoms is A318813.
Cf. A000669, A005121, A007716, A048816, A141268, A306186, A317791, A318812, A318849, A330470, A330475, A330655, A330728.

a(2^n) = a(prime(n)) = A318813(n).

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

			a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.
a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.
a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

A318563 Number of combinatory separations of strongly normal multisets of weight n.

Original entry on oeis.org

1, 4, 10, 33, 85, 272, 730, 2197, 6133

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, A318562, 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]]}];
Table[Length[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}]],{n,7}]

A318567 Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.

Original entry on oeis.org

1, 3, 8, 21, 54, 137, 343, 847, 2075, 5031, 12109, 28921, 68633, 161865, 379655

Offset: 1

Gus Wiseman, Aug 29 2018

Also the number of combinatory separations of normal multisets of weight n with constant parts. 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 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}.

			The a(3) = 8 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={1,11}
  112<={1,1,1}
  122<={1,11}
  122<={1,1,1}
  123<={1,1,1}

Cf. A000041, A007716, A011782, A034691, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318560, A318562, A318565.

Table[Sum[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@c]]],{c,Join@@Permutations/@IntegerPartitions[n]}],{n,30}]

cp's OEIS Frontend

A323794 Number of non-isomorphic weight-n multisets of sets of multisets.

Views

Author

Keywords

Comments

Examples

Crossrefs

A318287 Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A318369 Number of non-isomorphic set multipartitions (multisets of sets) of the multiset of prime indices of n.

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A318563 Number of combinatory separations of strongly normal multisets of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A318567 Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.

Views

Author

Keywords

Comments

Examples