A275780 -id:A275780 - cp's OEIS frontend

A038041 Number of ways to partition an n-set into subsets of equal size.

1, 2, 2, 5, 2, 27, 2, 142, 282, 1073, 2, 32034, 2, 136853, 1527528, 4661087, 2, 227932993, 2, 3689854456, 36278688162, 13749663293, 2, 14084955889019, 5194672859378, 7905858780927, 2977584150505252, 13422745388226152, 2, 1349877580746537123, 2

Offset: 1

Christian G. Bower

a(n) = 2 iff n is prime with a(p) = card{ 1|2|3|...|p-1|p, 123...p } = 2. - Bernard Schott, May 16 2019

			a(4) = card{ 1|2|3|4, 12|34, 14|23, 13|24, 1234 } = 5.
From _Gus Wiseman_, Jul 12 2019: (Start)
The a(6) = 27 set partitions:
  {{1}{2}{3}{4}{5}{6}}  {{12}{34}{56}}  {{123}{456}}  {{123456}}
                        {{12}{35}{46}}  {{124}{356}}
                        {{12}{36}{45}}  {{125}{346}}
                        {{13}{24}{56}}  {{126}{345}}
                        {{13}{25}{46}}  {{134}{256}}
                        {{13}{26}{45}}  {{135}{246}}
                        {{14}{23}{56}}  {{136}{245}}
                        {{14}{25}{36}}  {{145}{236}}
                        {{14}{26}{35}}  {{146}{235}}
                        {{15}{23}{46}}  {{156}{234}}
                        {{15}{24}{36}}
                        {{15}{26}{34}}
                        {{16}{23}{45}}
                        {{16}{24}{35}}
                        {{16}{25}{34}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..250
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A061095 (same but with labeled boxes), A005225, A236696, A055225, A262280, A262320.
Column k=1 of A208437.
Row sums of A200472 and A200473.
Cf. A000110, A007837 (different lengths), A035470 (equal sums), A275780, A317583, A320324, A322794, A326512 (equal averages), A326513.

A038041 := proc(n) local d;
add(n!/(d!*(n/d)!^d), d = numtheory[divisors](n)) end:
seq(A038041(n),n = 1..29); # Peter Luschny, Apr 16 2011

a[n_] := Block[{d = Divisors@ n}, Plus @@ (n!/(#! (n/#)!^#) & /@ d)]; Array[a, 29] (* Robert G. Wilson v, Apr 16 2011 *)
Table[Sum[n!/((n/d)!*(d!)^(n/d)), {d, Divisors[n]}], {n, 1, 31}] (* Emanuele Munarini, Jan 30 2014 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Length/@#&]],{n,0,8}] (* Gus Wiseman, Jul 12 2019 *)

a(n):= lsum(n!/((n/d)!*(d!)^(n/d)),d,listify(divisors(n)));
makelist(a(n),n,1,40); /* Emanuele Munarini, Feb 03 2014 */

/* compare to A061095 */
mnom(v)=
/* Multinomial coefficient s! / prod(j=1, n, v[j]!) where
  s= sum(j=1, n, v[j]) and n is the number of elements in v[]. */
sum(j=1, #v, v[j])! / prod(j=1, #v, v[j]!)
A038041(n)={local(r=0);fordiv(n,d,r+=mnom(vector(d,j,n/d))/d!);return(r);}
vector(33,n,A038041(n)) /* Joerg Arndt, Apr 16 2011 */

import math
def a(n):
    count = 0
    for k in range(1, n + 1):
        if n % k == 0:
            count += math.factorial(n) // (math.factorial(k) ** (n // k) * math.factorial(n // k))
    return count # Paul Muljadi, Sep 25 2024

a(n) = Sum_{d divides n} (n!/(d!*((n/d)!)^d)).

E.g.f.: Sum_{k >= 1} (exp(x^k/k!)-1).

More terms from Erich Friedman

A007837 Number of partitions of n-set with distinct block sizes.

Original entry on oeis.org

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847

Offset: 0

Arnold Knopfmacher

nonn

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..700
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011

nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)

{my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Christian G. Bower

a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A035470 Number of ways to break {1,2,3,...,n} into sets with equal sums.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 6, 12, 11, 2, 80, 166, 2, 665, 2918, 3309, 9296, 23730, 31875, 301030, 422897, 2, 13716867, 71504980, 100664385, 54148591, 880696662, 498017759, 27450476787, 111911522819, 179459955554, 2144502175214, 59115423983, 45837019664552, 375743493787258, 816118711787493, 2, 9492169507922

Offset: 1

Erich Friedman

nonn

a(n) = 2 <=> |{d|n*(n+1)/2 : d>=n}| = 2. - Alois P. Heinz, Sep 03 2009

			a(7) = 6 since we have 1234567, 16/25/34/7, 167/2345, 257/1346, 347/1256, 356/1247.
From _Gus Wiseman_, Jul 13 2019: (Start)
The a(6) = 2 through a(9) = 11 set partitions with equal block-sums:
  {123456}      {1234567}        {12345678}        {123456789}
  {16}{25}{34}  {1247}{356}      {12348}{567}      {12345}{69}{78}
                {1256}{347}      {12357}{468}      {1239}{456}{78}
                {1346}{257}      {12456}{378}      {1248}{357}{69}
                {167}{2345}      {1278}{3456}      {1257}{348}{69}
                {16}{25}{34}{7}  {1368}{2457}      {1347}{258}{69}
                                 {1458}{2367}      {1356}{249}{78}
                                 {1467}{2358}      {159}{2346}{78}
                                 {1236}{48}{57}    {159}{267}{348}
                                 {138}{246}{57}    {168}{249}{357}
                                 {156}{237}{48}    {18}{27}{36}{45}{9}
                                 {18}{27}{36}{45}
(End)

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

Cf. A164977, A164978. - Alois P. Heinz, Sep 03 2009
Row sums of A275714.
Row sums of A320438.
Cf. A000110, A007837, A038041, A112956, A275780, A275781, A321455, A326512, A326513, A326518, A326534.

with(numtheory): b:= proc() option remember; local i, j, t; `if`(args[1]=0, `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(`if`(args[j] -args[nargs] <0, 0, b(sort([seq(args[i] -`if`(i=j, args[nargs], 0), i=1..nargs-1)])[], args[nargs]-1)), j=1..nargs-1)) end: a:= proc(n) local i, m, x; m:= n*(n+1)/2; 1+ add(b(i$(m/i), n)/(m/i)!, i=[select(x-> x>=n, divisors(m) minus {m})[]]) end: seq(a(n), n=1..25);  # Alois P. Heinz, Sep 03 2009

b[args_List] := b[args] = If[args[[1]] == 0, If[Length[args] == 2, 1, b[Rest[args]]], Sum[If[args[[j]] - args[[-1]] < 0, 0, b[Sort[Join[Table[ args[[i]] - If[i == j, args[[-1]], 0], {i, 1, Length[args]-1}]]], {args[[-1]]-1}]], {j, 1, Length[args]-1}]]; b[a1_List, a2_List] := b[Join[a1, a2]];
a[n_] := a[n] = With[{m = n*(n+1)/2}, 1+Sum[b[Append[Array[i&, m/i], n]] / (m/i)!, {i, Select[Divisors[m] ~Complement~ {m}, # >= n &]}]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Mar 22 2017, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Total/@#&]],{n,0,10}] (* Gus Wiseman, Jul 13 2019 *)

More terms from John W. Layman, Mar 18 2002
a(19)-a(33) from Alois P. Heinz, Sep 03 2009
a(34) from Alois P. Heinz, May 24 2015
a(35)-a(38) from Max Alekseyev, Feb 15 2024

A279375 Number of set partitions of strict integer partitions of n that have distinct block-sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 12, 16, 24, 39, 49, 70, 94, 127, 202, 247, 340, 450, 606, 772, 1169, 1407, 1920, 2454, 3267, 4089, 5469, 7293, 9222, 11884, 15291, 19417, 24890, 31469, 39662, 52619, 64764, 82502, 103576, 131169, 162726, 206015, 254233, 318464, 406262, 499210, 620593, 773673, 957073, 1181593

Offset: 0

Gus Wiseman, Dec 11 2016

nonn

Also twice partitioned numbers where all partitions are strict. Also triangles of weight n in the multisystem of strict partitions. Strict partitions are an example of a multisystem that is neither transitive nor partitive nor contractible but is decomposable; see link for details.

			The a(6)=9 set partitions of strict integer partitions of 6 are: ((6)), ((51)), ((5)(1)), ((42)), ((4)(2)), ((321)), ((32)(1)), ((31)(2)), ((3)(2)(1)). The set partition ((3)(21)) is not counted because its blocks do not have distinct sums.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Gus Wiseman, Comcategories and Multiorders (pdf version)

Cf. A063834, A089259, A258466, A270995, A271619, A275780, A279374.

nn=20;sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Total[Length[Select[sps[Sort[#]],UnsameQ@@Total/@#&]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}]

A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 11 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 sum of prime indices of n is A056239(n).

			The a(72) = 8 multiset partitions with distinct block-sums:
    {{1,1,1,2,2}}
   {{1},{1,1,2,2}}
   {{2},{1,1,1,2}}
   {{1,1},{1,2,2}}
   {{1,2},{1,1,2}}
   {{2,2},{1,1,1}}
  {{1},{2},{1,1,2}}
  {{1},{1,1},{2,2}}
Missing from this list are:
    {{1},{1},{1,2,2}}
    {{1},{1,2},{1,2}}
    {{2},{2},{1,1,1}}
    {{2},{1,1},{1,2}}
   {{1},{1},{1},{2,2}}
   {{1},{1},{2},{1,2}}
   {{1},{2},{2},{1,1}}
  {{1},{1},{1},{2},{2}}

Cf. A001055, A056239, A275780, A317144, A321455, A322794, A326514, A326515, A326516, A326519, A326535, A371734.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]];
Table[Length[Select[mps[primeMS[n]],UnsameQ@@Sort[Total/@#]&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A321469(n, m=n, facs=List([])) = if(1==n, all_have_different_sum_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A321469(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

Original entry on oeis.org

1, 1, 2, 6, 23, 106, 549, 3184, 20353, 141615, 1063399, 8554800, 73281988, 665141182, 6369920854, 64133095134, 676690490875, 7462023572238, 85786458777923, 1025956348473929, 12739037494941490

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(3) = 6 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,2}}
                    {{2},{1,3}}
                    {{1},{2},{3}}
The a(4) = 23 factorizations:
  2*3*6  5*30    3*30    2*30    210
         10*15   6*15    6*10    2*105
         2*5*15  2*3*15  2*3*10  3*70
         3*5*10                  5*42
                                 7*30
                                 6*35
                                 10*21
                                 2*3*35
                                 2*5*21
                                 2*7*15
                                 3*5*14
                                 2*3*5*7

For distinct blocks instead of sums we have A116539, see A050326.
Without distinct sums we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279785.
Without strict blocks we have A326519.
Factorizations of this type are counted by A381633.
For constant instead of strict blocks we have A382203.
For distinct sizes instead of sums we have A382428, non-strict blocks A326517.
For equal instead of distinct block-sums we have A382429, non-strict blocks A326518.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A317532, A317583, A321469, A381635, A382204, A382214, A382216.

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[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(10)-a(11) from Robert Price, Mar 31 2025

a(12)-a(20) from Christian Sievers, Apr 05 2025

A326519 Number of normal multiset partitions of weight n where each part has a different sum.

Original entry on oeis.org

1, 1, 3, 11, 51, 259, 1461, 9133, 62348, 459547, 3632419

Offset: 0

Gus Wiseman, Jul 12 2019

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 11 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}
             {{1,2}}    {{1,1,2}}
             {{1},{2}}  {{1,2,2}}
                        {{1,2,3}}
                        {{1},{1,1}}
                        {{1},{1,2}}
                        {{1},{2,2}}
                        {{1},{2,3}}
                        {{2},{1,2}}
                        {{2},{1,3}}
                        {{1},{2},{3}}

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

Cf. A038041, A255906, A275780, A317583, A321469, A326517, A326518, A326520, A326521, A326535.

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],UnsameQ@@Total/@#&]],{n,0,5}]

a(8)-a(10) from Robert Price, Apr 03 2025

A326535 MM-numbers of multiset partitions where each part has a different sum.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A298540 in having 187.

These are numbers where each prime index has a different sum 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 each part has a different sum, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}

Robert Price, Table of n, a(n) for n = 1..9715
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A112798, A275780, A302242, A320324, A321469, A326519, A326533, A326534, A326536, A326537.

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

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

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

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

A326512 Number of set partitions of {1..n} where every block has the same average.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 18, 16, 75, 64, 405, 302, 2581, 1693, 19872, 11295, 175807, 87524, 1851135, 787515, 21909766, 8185713, 298698113, 96514608, 4538610230, 1285072142

Offset: 0

Gus Wiseman, Jul 11 2019

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024

			The a(1) = 1 through a(7) = 18 set partitions:
  {1}  {12}  {123}    {1234}    {12345}      {123456}      {1234567}
             {13}{2}  {14}{23}  {1245}{3}    {1256}{34}    {123567}{4}
                                {135}{24}    {1346}{25}    {12467}{35}
                                {15}{234}    {16}{2345}    {1267}{345}
                                {15}{24}{3}  {16}{25}{34}  {13457}{26}
                                                           {1357}{246}
                                                           {1456}{237}
                                                           {147}{2356}
                                                           {156}{2347}
                                                           {17}{23456}
                                                           {1267}{35}{4}
                                                           {1357}{26}{4}
                                                           {147}{26}{35}
                                                           {156}{237}{4}
                                                           {17}{2356}{4}
                                                           {17}{246}{35}
                                                           {17}{26}{345}
                                                           {17}{26}{35}{4}

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

Cf. A000110, A007837, A035470, A038041, A070925, A275780, A306017, A306021, A319169, A326513, A326515, A326520, A326536.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Mean/@#&]],{n,0,8}]

a(12)-a(15) from Alois P. Heinz, Jul 12 2019

a(16)-a(26) from Christian Sievers, Aug 22 2024

cp's OEIS Frontend

A038041 Number of ways to partition an n-set into subsets of equal size.

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A007837 Number of partitions of n-set with distinct block sizes.

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A035470 Number of ways to break {1,2,3,...,n} into sets with equal sums.

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A279375 Number of set partitions of strict integer partitions of n that have distinct block-sums.

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A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

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A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

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A326519 Number of normal multiset partitions of weight n where each part has a different sum.

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