A320664 -id:A320664 - cp's OEIS frontend

A320663 Number of non-isomorphic multiset partitions of weight n using singletons or pairs.

1, 1, 4, 7, 21, 40, 106, 216, 534, 1139, 2715, 5962, 14012, 31420, 73484, 167617, 392714, 908600, 2140429, 5015655, 11905145, 28228533, 67590229, 162067916, 391695348, 949359190, 2316618809, 5673557284, 13979155798, 34583650498, 86034613145, 214948212879

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}
         {{1,2}}    {{1},{2,2}}    {{1,1},{2,2}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{2}}  {{2},{1,2}}    {{1,2},{2,2}}
                    {{1},{1},{1}}  {{1,2},{3,3}}
                    {{1},{2},{2}}  {{1,2},{3,4}}
                    {{1},{2},{3}}  {{1,3},{2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

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

Cf. A001055, A001222, A001358, A005117, A006881, A007716, A007717, A037143, A320462, A320655, A320656, A320664, A320665.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((1+r)\2)*x^(2*r))}
a(n)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!} \\ Andrew Howroyd, Oct 26 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 26 2018

A330107 MM-numbers of brute-force normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359

Offset: 1

Gus Wiseman, Dec 02 2019

A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all brute-force normalized multiset partitions together with their MM-numbers begins:
   1: 0             63: {1}{1}{11}      159: {1}{1111}
   3: {1}           69: {1}{22}         161: {11}{22}
   7: {11}          81: {1}{1}{1}{1}    165: {1}{2}{3}
   9: {1}{1}        89: {1112}          169: {12}{12}
  13: {12}          91: {11}{12}        171: {1}{1}{111}
  15: {1}{2}       105: {1}{2}{11}      183: {1}{122}
  19: {111}        111: {1}{112}        189: {1}{1}{1}{11}
  21: {1}{11}      113: {123}           195: {1}{2}{12}
  27: {1}{1}{1}    117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         247: {12}{111}
  53: {1111}       147: {1}{11}{11}     259: {11}{112}
  57: {1}{111}     151: {1122}          267: {1}{1112}

Equals the odd terms of A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A300300, A316983, A317533, A320664, A330061, A330098, A330101, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

Original entry on oeis.org

1, 0, 1, 1, 5, 6, 27, 47, 169, 406, 1327, 3790, 12560, 39919, 136821, 470589, 1687981, 6162696, 23173374, 88981796, 349969596, 1405386733, 5764142220, 24111709328, 102825231702, 446665313598, 1975339030948, 8888051121242, 40667889052853, 189126710033882, 893526261542899

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

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}}. This sequence counts non-isomorphic multiset partitions with no singletons whose dual also has no singletons.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 27 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
                      {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
                      {{1,1},{1,1}}  {{1,1},{1,1,1}}  {{1,1,2,2,2,2}}
                      {{1,1},{2,2}}  {{1,1},{1,2,2}}  {{1,1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,1},{2,2,2}}  {{1,1},{1,1,1,1}}
                                     {{1,2},{1,2,2}}  {{1,1,1},{1,1,1}}
                                                      {{1,1},{1,2,2,2}}
                                                      {{1,1,1},{2,2,2}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,1},{2,2,2,2}}
                                                      {{1,1,2},{2,2,2}}
                                                      {{1,1},{2,2,3,3}}
                                                      {{1,1,2},{2,3,3}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,1},{1,2},{2,2}}
                                                      {{1,1},{2,2},{2,2}}
                                                      {{1,1},{2,2},{3,3}}
                                                      {{1,1},{2,3},{2,3}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,2},{2,2}}
                                                      {{1,2},{1,3},{2,3}}

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

Row sums of A369287.

Cf. A001055, A007716, A306005, A317794, A318871, A320663, A320664, A321760.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A-x*sv(1)), sExp(A-x*sv(1))))} \\ Andrew Howroyd, Jan 17 2023

Vec(G(20,1)) \\ G defined in A369287. - Andrew Howroyd, Jan 28 2024

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

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A320663 Number of non-isomorphic multiset partitions of weight n using singletons or pairs.

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A330107 MM-numbers of brute-force normalized multiset partitions.

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A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

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