A326533 -id:A326533 - cp's OEIS frontend

A326536 MM-numbers of multiset partitions where every part has the same average.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 133, 137, 139, 145, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A322902 in having 145.

These are numbers where each prime index has the same average of prime indices. 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 obtained by taking the multiset of prime indices of each prime index 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 multiset partitions where every part has the same average, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A051293, A112798, A302242, A320324, A326512, A326515, A326520, A326533, A326534, A326535, A326537.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Mean/@primeMS/@primeMS[#]&]

A326026 Number of non-isomorphic multiset partitions of weight n where each part has a different length.

Original entry on oeis.org

1, 1, 2, 7, 12, 35, 111, 247, 624, 1843, 6717, 15020, 46847, 124808, 412577, 1658973, 4217546, 12997734, 40786810, 126971940, 437063393, 2106317043, 5499108365, 19037901867, 59939925812, 210338815573, 683526043801, 2741350650705, 14848209030691, 41533835240731, 151548411269815

Offset: 0

Gus Wiseman, Jul 13 2019

nonn

The number of non-isomorphic multiset partitions of weight n is A007716(n).

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}
                  {{1},{2,2}}  {{1,2,3,4}}
                  {{1},{2,3}}  {{1},{1,1,1}}
                  {{2},{1,2}}  {{1},{1,2,2}}
                               {{1},{2,2,2}}
                               {{1},{2,3,3}}
                               {{1},{2,3,4}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

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

Row sums of A332260.

Cf. A007716, A007837, A303546, A306017, A316980, A316983, A319560, A326514, A326517, A326533.

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)); polcoef(prod(k=1, #u, 1 + u[k]*x^k + O(x*x^n)), n)/prod(i=1, #v, i^v[i]*v[i]!)}
a(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Feb 08 2020

Terms a(11) and beyond from Andrew Howroyd, Feb 08 2020

A358832 Number of twice-partitions of n into partitions of distinct lengths and distinct sums.

Original entry on oeis.org

1, 1, 2, 4, 7, 15, 25, 49, 79, 154, 248, 453, 748, 1305, 2125, 3702, 5931, 9990, 16415, 26844, 43246, 70947, 113653, 182314, 292897, 464614, 739640, 1169981, 1844511, 2888427, 4562850, 7079798, 11064182, 17158151, 26676385, 41075556, 63598025, 97420873, 150043132

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)      (4)       (5)
       (11)  (21)     (22)      (32)
             (111)    (31)      (41)
             (11)(1)  (211)     (221)
                      (1111)    (311)
                      (21)(1)   (2111)
                      (111)(1)  (11111)
                                (21)(2)
                                (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)

This is the case of A271619 with distinct lengths.
These multiset partitions are ranked by A326535 /\ A326533.
This is the case of A358830 with distinct sums.
For constant instead of distinct lengths and sums we have A358833.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A273873 counts strict trees.
Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A306319, A358334, A358836.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Total/@#&&UnsameQ@@Length/@#&]],{n,0,10}]

seq(n)={ local(Cache=Map());
  my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
  my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
  z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m],k)); if(!bittest(b,k)&&c, c*self()(min(m-1,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022

cp's OEIS Frontend

A326536 MM-numbers of multiset partitions where every part has the same average.

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A326026 Number of non-isomorphic multiset partitions of weight n where each part has a different length.

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A358832 Number of twice-partitions of n into partitions of distinct lengths and distinct sums.

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