A038041 -id:A038041 - cp's OEIS frontend

A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 4, 0, 7, 1, 9, 0, 16, 0, 21, 4, 32, 0, 45, 0, 63, 13, 84, 0, 126, 0, 158, 36, 220, 0, 303, 0, 393, 93, 511, 0, 708, 0, 881, 229, 1156, 0, 1539, 0, 1925, 516, 2445, 0, 3233, 6, 3952, 1134, 5019, 0, 6497

Offset: 0

Gus Wiseman, Apr 14 2024

nonn

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

Conjecture: (1) Positions of 0's are A327782. (2) Positions of terms > 0 are A368459.

			The a(0) = 0 through a(14) = 7 strict partitions:
  .  .  .  .  .  .  (321)  .  (431)  .  (532)   .  (642)   .  (743)
                                        (541)      (651)      (752)
                                        (4321)     (5421)     (761)
                                                   (6321)     (5432)
                                                              (6431)
                                                              (6521)
                                                              (7421)

The non-strict "bi-" version is A002219, ranks A357976.
The "bi-" version is A237258, ranks A357854, complement A321142 or A371794.
The non-strict version is A321452, ranks A321454.
The complement is A371736, non-strict A321451, ranks A321453.
The non-strict "bi-" complement is A371795, ranks A371731.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, complement A371792.
A371796 counts quanimous sets, complement A371789.
Cf. A000005, A018818, A035470, A038041, A064688, A232466, A237194, A305551, A365663, A365661, A365925, A371733, A371839.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Select[sps[#], SameQ@@Total/@#&]]>1&]],{n,0,30}]

A371797 Number of quanimous subsets of {1..n} containing n, meaning there is more than one set partition with equal block-sums.

Original entry on oeis.org

0, 0, 1, 2, 5, 11, 24, 51, 112, 233, 507, 1044, 2214, 4557, 9472, 19545, 40373, 82145, 168374, 341523, 693350, 1408893, 2860365, 5771355, 11667351, 23542022, 47484577, 95861243, 193447849, 389602553

Offset: 1

Gus Wiseman, Apr 17 2024

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is counted under a(9).
The a(1) = 0 through a(6) = 11 subsets:
  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                 {1,2,3,4}  {2,3,5}      {2,4,6}
                            {1,2,4,5}    {1,2,3,6}
                            {2,3,4,5}    {1,2,5,6}
                            {1,2,3,4,5}  {1,3,4,6}
                                         {2,3,5,6}
                                         {3,4,5,6}
                                         {1,2,3,4,6}
                                         {1,2,4,5,6}
                                         {2,3,4,5,6}
                                         {1,2,3,4,5,6}

The "bi-" version is A232466, complement A371793.
The complement is counted by A371790.
First differences of A371796, complement A371789.
A371736 counts non-quanimous strict partitions.
A371737 counts quanimous strict partitions.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous subsets, complement A371792.
Cf. A000005, A002219, A035470, A038041, A275972, A279791, A321452, A371795.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,10}]

a(11)-a(30) from Martin Fuller, Apr 01 2025

A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

Original entry on oeis.org

1, 2, 3, 13, 25, 301, 721, 10921, 60481, 740881, 3628801, 106777441, 479001601, 12462690241, 134399865601, 2833553923201, 20922789888001, 892191453753601, 6402373705728001, 268633265290790401, 3652732042831872001, 102181898422712908801, 1124000727777607680001

Offset: 1

Vladeta Jovovic, Oct 15 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..450

Cf. A057625, A061095, A038041, A038048, A005225, A209902 (exp).

Array[n \[Function] DivisorSum[n, (n - 1)!/(# - 1)! &], 25] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

a(n)=sumdiv(n,d,(n-1)!/(d-1)!); \\ Joerg Arndt, May 21 2013

E.g.f.: Sum_{k>0} (exp(x^k)-1)/k = -Sum_{k>0} log(1-x^k)/k!.

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A326513 Number of set partitions of {1..n} where each block has a different average.

Original entry on oeis.org

1, 1, 2, 4, 12, 40, 154, 650, 3024, 15110, 81538, 468494, 2863340, 18481390, 125838194, 897725927, 6715102246, 52372397021, 425716871241, 3594451206166, 31509992921241, 285799247349838, 2682935185643622, 25990339824995969, 259777696236210943, 2673388551328088666

Offset: 0

Gus Wiseman, Jul 11 2019

nonn

			The a(1) = 1 through a(4) = 12 set partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                    {{1,2},{3}}    {{1,2},{3,4}}
                    {{1},{2},{3}}  {{1,2,3},{4}}
                                   {{1,2,4},{3}}
                                   {{1,3},{2,4}}
                                   {{1,3,4},{2}}
                                   {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

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

Cf. A000110, A007837, A035470, A038041, A275780, A326512, A326516, A326521, A326537.

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

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

a(13)-a(25) from Christian Sievers, Aug 20 2024

A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 12 2019

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 a(60) = 8 factorizations: (2*5*6), (3*4*5), (2*30), (3*20), (4*15), (5*12), (6*10), (60).

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326513, A326514, A326515, A326521, A326537.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@Mean/@primeMS/@#&]],{n,100}]

avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
all_have_different_average_of_pis(facs) = if(!#facs, 1, (#Set(apply(avgpis,facs)) == #facs));
A326516(n, m=n, facs=List([])) = if(1==n, all_have_different_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326516(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

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

A371790 Number of non-quanimous subsets of {1..n} containing n, meaning there is only one set partition with equal block-sums.

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 40, 77, 144, 279, 517, 1004, 1882, 3635, 6912, 13223, 25163, 48927, 93770, 182765, 355226, 688259, 1333939, 2617253, 5109865, 10012410, 19624287, 38356485, 74987607, 147268359

Offset: 1

Gus Wiseman, Apr 17 2024

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is not counted under a(9).
The a(1) = 1 through a(5) = 11 subsets:
  {1}  {2}    {3}    {4}      {5}
       {1,2}  {1,3}  {1,4}    {1,5}
              {2,3}  {2,4}    {2,5}
                     {3,4}    {3,5}
                     {1,2,4}  {4,5}
                     {2,3,4}  {1,2,5}
                              {1,3,5}
                              {2,4,5}
                              {3,4,5}
                              {1,2,3,5}
                              {1,3,4,5}

First differences of A371789, complement counted by A371796.
The "bi-" version is A371793, complement A232466.
The complement is counted by A371797.
A371736 counts non-quanimous strict partitions.
A371737 counts quanimous strict partitions.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous subsets, complement A371792.
Cf. A002219, A035470, A038041, A275972, A316402, A321451, A321452.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Length[Select[sps[#],SameQ@@Total/@#&]]==1&]],{n,10}]

a(11)-a(30) from Martin Fuller, Apr 01 2025

A371736 Number of non-quanimous strict integer partitions of n, meaning no set partition with more than one block has all equal block-sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 26, 23, 38, 30, 54, 43, 72, 57, 104, 77, 142, 102, 179, 138, 256, 170, 340, 232, 412, 292, 585, 365, 760, 471, 889, 602, 1260, 718, 1610, 935, 1819, 1148, 2590, 1371, 3264, 1733, 3581, 2137, 5120, 2485, 6372

Offset: 0

Gus Wiseman, Apr 14 2024

nonn

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

			The a(0) = 1 through a(9) = 8 strict partitions:
  ()  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
                (21)  (31)  (32)  (42)  (43)   (53)   (54)
                            (41)  (51)  (52)   (62)   (63)
                                        (61)   (71)   (72)
                                        (421)  (521)  (81)
                                                      (432)
                                                      (531)
                                                      (621)

The non-strict "bi-" complement is A002219, ranks A357976.
The "bi-" version is A321142 or A371794, complement A237258, ranks A357854.
The non-strict version is A321451, ranks A321453.
The complement is A371737, non-strict A321452, ranks A321454.
The non-strict "bi-" version is A371795, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371792 counts non-biquanimous sets, complement A371791.
A371796 counts quanimous sets, differences A371797.
Cf. A000005, A018818, A035470, A038041, A232466, A279791, A365663, A365661, A365925, A371840.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Select[sps[#], SameQ@@Total/@#&]]==1&]],{n,0,30}]

a(prime(k)) = A064688(k) = A000009(A000040(k)).

A299353 Number of labeled connected uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 5, 50, 1713, 1101990, 68715891672, 1180735735356264714926, 170141183460507906731293351306487161569, 7237005577335553223087828975127304177495735363998991435497132228228565768846

Offset: 0

Gus Wiseman, Jun 18 2018

nonn

A hypergraph is uniform if all edges have the same size.

Let T be the regular triangle A299354, where column k is the logarithmic transform of the inverse binomial transform of c(d) = 2^binomial(d,k). Then a(n) is the sum of row n.

			The a(3) = 5 hypergraphs:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}

Wikipedia, Hypergraph

Cf. A000005, A001315, A006126, A006129, A038041, A048143, A298422, A299354, A299471, A306020, A306021.

nn=10;Table[Sum[SeriesCoefficient[Log[Sum[x^m/m!*(-1)^(m-d)*Binomial[m,d]*2^Binomial[d,k],{m,0,n},{d,0,m}]],{x,0,n}]*n!,{k,n}],{n,nn}]

A322451 Number of unlabeled 3-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 29, 2102, 7011184, 1788775603336, 53304526022885280592, 366299663378889804782337225824, 1171638318502622784366970315264281830913536, 3517726593606524901243694560022510194223171115509135178240

Offset: 0

Gus Wiseman, Dec 09 2018

nonn

3-uniform means that every edge consists of 3 vertices. - Brendan McKay, Sep 03 2023

			Non-isomorphic representatives of the a(5) = 29 hypergraphs:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{123}{145}{245}{345}}
  {{124}{135}{245}{345}}
  {{125}{135}{245}{345}}
  {{134}{235}{245}{345}}
  {{145}{235}{245}{345}}
  {{123}{124}{135}{245}{345}}
  {{123}{145}{235}{245}{345}}
  {{124}{134}{235}{245}{345}}
  {{134}{145}{235}{245}{345}}
  {{135}{145}{235}{245}{345}}
  {{145}{234}{235}{245}{345}}
  {{123}{124}{134}{235}{245}{345}}
  {{123}{134}{145}{235}{245}{345}}
  {{123}{145}{234}{235}{245}{345}}
  {{124}{135}{145}{235}{245}{345}}
  {{125}{135}{145}{235}{245}{345}}
  {{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{235}{245}{345}}
  {{124}{135}{145}{234}{235}{245}{345}}
  {{125}{135}{145}{234}{235}{245}{345}}
  {{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{234}{235}{245}{345}}
  {{125}{134}{135}{145}{234}{235}{245}{345}}
  {{124}{125}{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}

Andrew Howroyd, Table of n, a(n) for n = 0..25

First differences of A000665.

Cf. A006126, A006129, A038041, A299471, A301922, A302374, A302394, A306017, A306021.

a(12) from Andrew Howroyd, Dec 15 2018

Name corrected by Brendan McKay, Sep 03 2023

A326514 Number of factorizations of n into factors > 1 where each factor has a different number of prime factors counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 5, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 3, 4, 1, 4, 1, 3, 1, 4, 1, 6, 1, 1, 3, 3, 1, 4, 1, 5, 2, 1, 1, 4, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 11 2019

nonn

			The a(96) = 8 factorizations: (2*4*12), (2*6*8), (2*48), (3*4*8), (3*32), (4*24), (6*16), (96).

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

Cf. A001055, A007837, A038041, A321455, A321469, A322794, A326026, A326515, A326516, A326517, A326533.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@PrimeOmega/@#&]],{n,100}]

cp's OEIS Frontend

A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.

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A371797 Number of quanimous subsets of {1..n} containing n, meaning there is more than one set partition with equal block-sums.

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A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

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A326513 Number of set partitions of {1..n} where each block has a different average.

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A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

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A371790 Number of non-quanimous subsets of {1..n} containing n, meaning there is only one set partition with equal block-sums.

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A371736 Number of non-quanimous strict integer partitions of n, meaning no set partition with more than one block has all equal block-sums.

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A299353 Number of labeled connected uniform hypergraphs spanning n vertices.

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