A255906 -id:A255906 - cp's OEIS frontend

A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.

0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 86, 87, 92, 93, 94, 95, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 0

Gus Wiseman, Mar 27 2024

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 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.

			The terms together with their prime indices of binary indices begin:
   0: {}
   1: {{}}
   2: {{1}}
   3: {{},{1}}
   6: {{1},{2}}
   7: {{},{1},{2}}
   8: {{1,1}}
   9: {{},{1,1}}
  10: {{1},{1,1}}
  11: {{},{1},{1,1}}
  12: {{2},{1,1}}
  13: {{},{2},{1,1}}
  14: {{1},{2},{1,1}}
  15: {{},{1},{2},{1,1}}
  22: {{1},{2},{3}}
  23: {{},{1},{2},{3}}
  28: {{2},{1,1},{3}}
  29: {{},{2},{1,1},{3}}
  30: {{1},{2},{1,1},{3}}
  31: {{},{1},{2},{1,1},{3}}
  32: {{1,2}}

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

The case with squarefree product of prime indices is A371293.
For binary indices of each prime index we have A371447, A371448.
The connected components of this multiset system are counted by A371452.
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, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A001222, A055887, A255906, A326782, A368109, A371291, A371294.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],normQ[Join@@prix/@bpe[#]]&]

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    for n in count(0):
        s = set()
        b = [(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1']
        for i in b:
            p = factorint(i)
            for j in p:
                s.add(sieve.search(j)[0])
        x = sorted(s)
        y = len(x)
        if sum(x) == (y*(y+1))//2:
            yield n
A371292_list = list(islice(a_gen(), 65)) # John Tyler Rascoe, May 21 2024

A317655 Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 3, 8, 10, 15, 50, 35, 37, 96, 144, 160, 299, 184, 589, 840, 2483, 578, 1729, 750, 10746, 1627, 2246, 3578, 9357, 3367, 47420, 6397, 212668, 3155, 9818, 17280, 15666, 18250, 966324, 84232, 54990, 12471, 4439540, 45015

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(6) = 8 free pure symmetric multifunctions:
  1[1[2]]
  1[2[1]]
  2[1[1]]
  1[1][2]
  1[2][1]
  2[1][1]
  1[1,2]
  2[1,1]

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317656, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[exprUsing[got[Reverse[primeMS[n]]]]],{n,40}]

A317656 Number of free pure symmetric multifunctions whose leaves are the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 2, 10, 1, 8, 1, 8, 2, 2, 1, 35, 1, 2, 3, 8, 1, 15, 1, 37, 2, 2, 2, 50, 1, 2, 2, 35, 1, 15, 1, 8, 8, 2, 1, 160, 1, 8, 2, 8, 1, 35, 2, 35, 2, 2, 1, 96, 1, 2, 8, 144, 2, 15, 1, 8, 2, 15, 1, 299, 1, 2, 8, 8, 2, 15, 1, 160

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.

			The a(12) = 8 free pure symmetric multifunctions are 1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1].

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317658.

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]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0,{},If[Length[m]==1,{First[m]},Join@@Cases[Union[Table[PR[m[[s]],m[[Complement[Range[Length[m]],s]]]],{s,Take[Subsets[Range[Length[m]]],{2,-2}]}]],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h],Union[Sort/@Tuples[exprUsing/@p]]}],{p,mps[g]}]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[exprUsing[primeMS[n]]],{n,100}]

A330783 Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 3, 8, 27, 94, 385, 1673, 8079, 41614, 231447, 1364697, 8559575, 56544465, 393485452, 2867908008, 21869757215, 173848026202, 1438593095272, 12360614782433, 110119783919367, 1015289796603359, 9674959683612989, 95147388659652754, 964559157655032720, 10067421615492769230

Offset: 0

Gus Wiseman, Jan 02 2020

nonn

The (weakly) normal version is A116540.

			The a(1) = 1 through a(3) = 8 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
The a(4) = 27 set multipartitions:
  {{1},{1},{1},{1}}  {{1},{1},{1,2}}  {{1},{1,2,3}}  {{1,2,3,4}}
  {{1},{1},{1},{2}}  {{1},{1},{2,3}}  {{1,2},{1,2}}
  {{1},{1},{2},{2}}  {{1},{2},{1,2}}  {{1,2},{1,3}}
  {{1},{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{2,3,4}}
  {{1},{2},{3},{4}}  {{1},{2},{3,4}}  {{1,2},{3,4}}
                     {{1},{3},{1,2}}  {{1,3},{2,4}}
                     {{1},{3},{2,4}}  {{1,4},{2,3}}
                     {{1},{4},{2,3}}  {{2},{1,3,4}}
                     {{2},{3},{1,4}}  {{3},{1,2,4}}
                     {{2},{4},{1,3}}  {{4},{1,2,3}}
                     {{3},{4},{1,2}}

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

Allowing edges to be multisets gives is A035310.
The strict case is A318402.
The constant case is A000005.
The (weakly) normal version is A116540.
Unlabeled set multipartitions are A049311.
Set multipartitions of prime indices are A050320.
Set multipartitions of integer partitions are A089259.
Cf. A001055, A047968, A255906, A269134, A283877, A296119, A317775, A318360, A318362, A330625, A330628.

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];
Table[Length[Select[Join@@mps/@strnorm[n],And@@UnsameQ@@@#&]],{n,0,5}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n)))/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A356937 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 9, 29, 94, 310, 1026, 3411, 11360, 37886, 126442, 422203, 1410189, 4711039, 15740098, 52593430, 175742438, 587266782, 1962469721, 6558071499, 21915580437, 73237274083, 244744474601, 817889464220, 2733235019732, 9133973730633, 30524096110942, 102006076541264

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

An interval such as {3,4,5} is a set with all differences of adjacent elements equal to 1.

			The a(1) = 1 through a(3) = 9 set multipartitions (multisets of sets):
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A055887, A063834, A270995, A304969, A349050, A349055.
Intervals are counted by A000012, A001227, ranked by A073485.
Other conditions: A034691, A116540, A255906, A356933, A356942.
Other types: A107742, A356936, A356938, A356939.

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[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@allnorm[n],And@@chQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, max(0, 1+k-j)))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.

Original entry on oeis.org

1, 1, 4, 15, 61, 249, 1040, 4363, 18424, 78014, 331099, 1407080, 5985505, 25477399, 108493103, 462147381, 1969025286, 8390475609, 35757524184, 152398429323, 649555719160, 2768653475487, 11801369554033, 50304231997727, 214428538858889, 914039405714237

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

			The a(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,2}}
         {{1},{2}}  {{1,2,3}}
                    {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A063834, A072233, A270995, A304969, A349050, A349055, A356934.
Gapless multisets are counted by A034296, ranked by A073491.
Other conditions: A034691, A055887, A116540, A255906, A356933, A356937.
Other types of multiset partitions: A356233, A356941, A356943, A356944.

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]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@allnorm[n],And@@nogapQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, sum(i=1, min(k, j), (k-i+1)*binomial(j-1, i-1))))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023

A303551 Number of aperiodic multisets of compositions of total weight n.

Original entry on oeis.org

1, 2, 6, 15, 41, 95, 243, 567, 1366, 3189, 7532, 17428, 40590, 93465, 215331, 493150, 1127978, 2569049, 5841442, 13240351, 29953601, 67596500, 152258270, 342235866, 767895382, 1719813753, 3845442485, 8584197657, 19133459138, 42583565928, 94641591888

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 15 aperiodic multisets of compositions are:
{4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},
{1,3}, {1,21}, {1,12}, {1,111}, {2,11},
{1,1,2}, {1,1,11}.
Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303546, A303552.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..35);  # Alois P. Heinz, Apr 26 2018

nn=20;
ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=EulerT(vector(n, n, 2^(n-1)))); vector(n, n, sumdiv(n, d, moebius(d)*u[n/d]))} \\ Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} mu(d) * A034691(n/d).

A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 5, 21, 16, 5, 7, 52, 72, 32, 7, 11, 141, 306, 216, 65, 11, 15, 327, 1113, 1160, 512, 113, 15, 22, 791, 4033, 6052, 3737, 1154, 199, 22, 30, 1780, 13586, 28749, 24325, 10059, 2317, 323, 30, 42, 4058, 45514, 133642, 151994, 82994, 24854, 4493, 523, 42

Offset: 1

Gus Wiseman, Aug 06 2018

			The T(3,2) = 6 multiset partitions are {{1},{1,1}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
Triangle begins:
    1
    2    2
    3    6    3
    5   21   16    5
    7   52   72   32    7
   11  141  306  216   65   11
   15  327 1113 1160  512  113   15
   ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A035310. First and last columns are both A000041.

Cf. A001055, A007716, A045778, A255906, A281116, A317584, A317654, A317755, A317775, A317776.

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];
Table[Length[Select[Join@@mps/@strnorm[n],Length[#]==k&]],{n,6},{k,n}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
U(m, n)={my(s=0); forpart(p=m, s+=D(p,n)); s}
M(n)={Mat(vector(n,k,(U(k,n)-U(k-1,n))~))}
{ my(A=M(8)); for(n=1, #A~, print(A[n,1..n])) } \\ Andrew Howroyd, Dec 30 2020

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2020

A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

Original entry on oeis.org

1, 1, 2, 8, 28, 108, 524, 2608, 14176, 86576, 550672, 3782496, 27843880, 214071392, 1751823600, 15041687664, 134843207240, 1269731540864, 12427331494304, 126619822952928, 1341762163389920, 14712726577081248, 167209881188545344, 1963715680476759040, 23794190474350155856

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

			The a(4) = 28 multiset partitions:
  {1}{111}      {1}{112}      {1}{123}      {1}{234}
  {1}{1}{1}{1}  {1}{122}      {1}{223}      {2}{134}
                {1}{222}      {1}{233}      {3}{124}
                {2}{111}      {2}{113}      {4}{123}
                {2}{112}      {2}{123}      {1}{2}{3}{4}
                {2}{122}      {2}{133}
                {1}{1}{1}{2}  {3}{112}
                {1}{1}{2}{2}  {3}{122}
                {1}{2}{2}{2}  {3}{123}
                              {1}{1}{2}{3}
                              {1}{2}{2}{3}
                              {1}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 0..500
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A055887, A063834, A072233, A270995, A304969, A349050, A349055.
Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other conditions: A034691, A116540, A255906, A356937, A356942.
Other types: A050330, A356932, A356934, A356935.

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]];
Table[Length[Select[Join@@mps/@allnorm[n],OddQ[Times@@Length/@#]&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, if(j%2 == 1, binomial(j+k-1, j))))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Jan 01 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

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 m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {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}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

allnorm[n_Integer]:=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[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]

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