A038041 -id:A038041 - cp's OEIS frontend

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1

Offset: 0

Gus Wiseman, Apr 16 2024

			The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
     1
     0     1
     0     1     1
     0     2     2     1
     0     2     8     4     1
     0     2    19    24     6     1
     0     2    47    95    49     9     1
     0     6   105   363   297    93    12     1
     0    12   248  1292  1660   753   158    16     1
     0    11   563  4649  8409  5591  1653   250    20     1
     0     2  1414 15976 41264 38074 15590  3249   380    25     1
Row n = 4 counts the following set partitions:
  .  {{1,4},{2,3}}  {{1},{2,3,4}}    {{1},{2},{3,4}}  {{1},{2},{3},{4}}
     {{1,2,3,4}}    {{1,2},{3},{4}}  {{1},{2,3},{4}}
                    {{1,2},{3,4}}    {{1},{2,4},{3}}
                    {{1,3},{2},{4}}  {{1,4},{2},{3}}
                    {{1,3},{2,4}}
                    {{1,2,3},{4}}
                    {{1,2,4},{3}}
                    {{1,3,4},{2}}

Row sums are A000110.
Column k = 1 is A035470.
A version for integer partitions is A116608.
For block lengths instead of sums we have A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
A371737 counts quanimous strict partitions, non-strict A321452.
A371789 counts non-quanimous sets, differences A371790.
Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]

A200472 Triangle T(n,k) is the number of ways to assign n people to k unlabeled groups of equal size.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 0, 1, 1, 0, 0, 0, 1, 1, 10, 15, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 35, 0, 105, 0, 0, 0, 1, 1, 0, 280, 0, 0, 0, 0, 0, 1, 1, 126, 0, 0, 945, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 462, 5775, 15400, 0, 10395, 0, 0, 0, 0, 0, 1

Offset: 1

Dennis P. Walsh, Nov 18 2011

If k is not a factor of n, T(n,k) = 0. If k is a factor of n, T(n,k) = (n!/k!)/(n/k)!^k. If n is a multiple of k, we may obtain T(n,k) by arranging all n people in an ordered line, which can be done in n! ways. Peel off the first n/k people for "group 1", the next n/k people for "group 2", ..., and the last n/k people for "group k". Since the k groups are actually unlabeled, we must divide n! by k! Also, since the ordering of the n/k people within each of the k groups is not of importance, we must now divide by (n/k)!^k. Therefore, T(n,k) = (n!/k!)/(n/k)!^k.

Also, T(2n,n) provide the sequence consisting of the products of consecutive odd integers.

			Triangle T(n,k) begins
1;
1,   1;
1,   0,    1;
1,   3,    0,     1;
1,   0,    0,     0,   1;
1,  10,   15,     0,   0,     1;
1,   0,    0,     0,   0,     0,  1;
1,  35,    0,   105,   0,     0,  0,  1;
1,   0,  280,     0,   0,     0,  0,  0,  1;
1, 126,    0,     0, 945,     0,  0,  0,  0,  1;
1,   0,    0,     0,   0,     0,  0,  0,  0,  0,  1;
1, 462, 5775, 15400,   0, 10395,  0,  0,  0,  0,  0,  1;
...
T(6,2) = 10 since there are 10 ways to assign 6 people (A,B,C,D,E,F) into 2 groups of size 3. The assignments are {A,B,C}|{D,E,F}, {A,B,D}|{C,E,F}, {A,B,E}|{C,D,F}, {A,B,F}|{C,D,E}, {A,C,D}|{B,E,F}, {A,C,E}|{B,D,F}, {A,C,F}|{B,D,E}, {B,C,D}|{A,E,F}, {B,C,E}|{A,D,F}, and {B,C,F}|{A,D,E}.

Dennis P. Walsh, Note on assigning n people into k unlabeled groups of equal size

T(2n,n) is A001147(n).
T(3n,n) is A025035(n).
T(4n,n) is A025036(n).
Row sums of T(n,k) provide A038041(n).
A200473 is A200472 with zeros removed.

T:= (n, k)-> `if`(modp(n, k)=0, n!/(k!*((n/k)!)^k), 0):
seq(seq(T(n, k), k=1..n), n=1..20);

nn=11;s=Sum[Exp[y x^i/i!]-1,{i,1,nn}];Range[0,nn]!CoefficientList[Series[s,{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Sep 15 2012 *)

T(n,k) = if(n%k!=0, 0, (n!/k!)/((n/k)!)^k );
for (n=1,15, for (k=1,n, print1(T(n,k),", "));print());
/* Joerg Arndt, Sep 16 2012 */

For k that divide n, T(n,k) = (n!/k!)/((n/k)!)^k; otherwise, T(n,k) = 0.
E.g.f. when k is fixed: (1/k!) sum(j>=1, (x^j/j!)^k ).
E.g.f. for T(n*r,n): exp(x^r/r!).
T(2n,n) = (2n-1)!! = (2n-1)(2n-3)...(3)(1).

A262320 Number of ways to select a subset s from an n-set and then partition s into blocks of equal size.

Original entry on oeis.org

1, 2, 5, 12, 30, 73, 191, 528, 1553, 5032, 18088, 66905, 266382, 1164517, 5215645, 23868104, 117740144, 609872351, 3268548407, 18110463456, 102867877415, 620476915966, 4005216028162, 25747549921339, 166978155172421, 1168774024335204, 8556355097320142

Offset: 0

Alois P. Heinz, Sep 17 2015

nonn

			a(3) = 12: {}, 1, 2, 3, 12, 1|2, 13, 1|3, 23, 2|3, 123, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..616

Partial sums of A262321.

Cf. A038041, A262280.

b:= proc(n) option remember;
      add(1/(d!*(n/d)!^d), d=numtheory[divisors](n))
    end:
a:= n-> 1 + n! * add(b(k)/(n-k)!, k=1..n):
seq(a(n), n=0..30);

b[n_] := b[n] = DivisorSum[n, 1/(#!*(n/#)!^#)&]; a[n_] := 1 + n! * Sum[b[k]/(n-k)!, {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

E.g.f.: exp(x) * (1 + Sum_{k>=1} (exp(x^k/k!)-1)).
a(n) = 1 + Sum_{k=1..n} C(n,k) * A038041(k).
a(n) = 1 + A262280(n).
a(n) = Sum_{k=0..n} A262321(k).

A299354 Regular triangle where T(n,k) is the number of labeled connected k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 38, 11, 1, 0, 728, 958, 26, 1, 0, 26704, 1042632, 32596, 57, 1, 0, 1866256, 34352418950, 34359509614, 2096731, 120, 1, 0, 251548592, 72057319189266922, 1180591620442534312262, 72057594021152435, 268434467, 247, 1, 0, 66296291072

Offset: 1

Gus Wiseman, Jun 18 2018

			Triangle begins:
1
0, 1
0, 4, 1
0, 38, 11, 1
0, 728, 958, 26, 1
0, 26704, 1042632, 32596, 57, 1

Wikipedia, Hypergraph

Second column is A001187. Row sums are A299353.

Cf. A000005, A001315, A006126, A006129, A038041, A048143, A298422, A299353, A299471, A306020, A306021.

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

Column k is the logarithmic transform of the inverse binomial transform of c(d) = 2^binomial(d,k).

A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 6, 19, 14, 113, 30, 584, 1150, 4023, 112, 119866, 202, 432061, 5442765, 16646712, 594, 738090160, 980, 13160013662, 113864783987, 39049423043, 2510, 44452496723053, 19373518220009, 21970704599961, 8858890258339122, 43233899006497146, 9130, 4019875470540832643

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(4) = 19 multiset partitions:
  {{1,1,1,1}}, {{1,1},{1,1}}, {{1},{1},{1},{1}},
  {{1,1,1,2}}, {{1,1},{1,2}}, {{1},{1},{1},{2}},
  {{1,1,2,2}}, {{1,1},{2,2}}, {{1,2},{1,2}}, {{1},{1},{2},{2}},
  {{1,1,2,3}}, {{1,1},{2,3}}, {{1,2},{1,3}}, {{1},{1},{2},{3}},
  {{1,2,3,4}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}}, {{1},{2},{3},{4}}.

Cf. A000005, A000041, A007716, A038041, A255906, A298422, A306017, A306018, A306019, A306020, A306021, A317583.

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],SameQ@@Length/@#&]],{n,6}]

\\ See links in A339645 for combinatorial species functions.
cycleIndex(n)={sum(n=1, n, x^n*sumdiv(n, d, sApplyCI(symGroupCycleIndex(d), d, symGroupCycleIndex(n/d), n/d))) + O(x*x^n)}
StronglyNormalLabelingsSeq(cycleIndex(15)) \\ Andrew Howroyd, Jan 01 2021

a(p) = 2*A000041(p) for prime p. - Andrew Howroyd, Jan 01 2021

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2021

A319540 Number of unlabeled 3-uniform hypergraphs spanning n vertices such that every pair of vertices appears together in some block.

Original entry on oeis.org

1, 1, 0, 1, 2, 14, 964, 3908438

Offset: 0

Gus Wiseman, Jan 09 2019

			Non-isomorphic representatives of the a(5) = 14 hypergraphs:
              {{123}{145}{245}{345}}
            {{123}{124}{135}{245}{345}}
            {{123}{145}{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}}
        {{123}{124}{135}{145}{235}{245}{345}}
        {{124}{135}{145}{234}{235}{245}{345}}
        {{125}{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}}

The labeled case is A302394.

Cf. A000665, A006126, A006129, A038041, A299471, A301922, A302374, A306021, A320395, A322451.

a(6)-a(7) from Andrew Howroyd, Aug 17 2019

A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 2, 2, 4, 2, 3, 3, 4, 4, 4, 3, 5, 4, 5, 6, 6, 6, 6, 6, 8, 6, 7, 9, 8, 11, 8, 11, 11, 11, 12, 13, 13, 15, 13, 17, 17, 18, 18, 17, 20, 22, 21, 24, 24, 24, 26, 29, 28, 33, 30, 35, 34, 38, 38, 45, 42, 43, 45, 48, 52, 54, 55, 59, 59, 65, 65, 72, 73

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.

			The a(30) = 8 integer partitions:
  (30),
  (17,13),(19,11),(23,7),
  (17,11,2),(23,5,2),
  (13,7,5,3,2),
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Lucas A. Brown, Table of n, a(n) for n = 0..133
Lucas A. Brown, Python program.

Cf. A002865, A003963, A005117, A038041, A064573, A073576, A302505, A319056, A320322, A321717, A321718, A322526, A322527, A322528, A322530, A322531.

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]

a(51)-a(69) from Jinyuan Wang, Jun 27 2020

a(70) onwards from Lucas A. Brown, Aug 17 2024

A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 2, 2, 5, 4, 3, 2, 2, 27, 11, 6, 4, 2, 2, 142, 29, 8, 4, 282, 12, 3, 1073, 101, 8, 4, 2, 2, 32034, 1581, 234, 75, 20, 6, 2, 2, 136853, 2660, 10, 4, 1527528, 1985, 91, 4, 4661087, 64596, 648, 20, 5, 2, 2, 227932993, 1280333, 41945, 231, 28, 6

Offset: 1

Gus Wiseman, Dec 26 2018

A multiset partition is uniform if all parts have the same size.

			Triangle begins:
     1
     2    2
     2    2
     5    4    3
     2    2
    27   11    6    4
     2    2
   142   29    8    4
   282   12    3
  1073  101    8    4
The multiset partitions counted under row 6:
  {123456}          {112233}          {111222}          {111111}
  {123}{456}        {112}{233}        {111}{222}        {111}{111}
  {124}{356}        {113}{223}        {112}{122}        {11}{11}{11}
  {125}{346}        {122}{133}        {11}{12}{22}      {1}{1}{1}{1}{1}{1}
  {126}{345}        {123}{123}        {12}{12}{12}
  {134}{256}        {11}{22}{33}      {1}{1}{1}{2}{2}{2}
  {135}{246}        {11}{23}{23}
  {136}{245}        {12}{12}{33}
  {145}{236}        {12}{13}{23}
  {146}{235}        {13}{13}{22}
  {156}{234}        {1}{1}{2}{2}{3}{3}
  {12}{34}{56}
  {12}{35}{46}
  {12}{36}{45}
  {13}{24}{56}
  {13}{25}{46}
  {13}{26}{45}
  {14}{23}{56}
  {14}{25}{36}
  {14}{26}{35}
  {15}{23}{46}
  {15}{24}{36}
  {15}{26}{34}
  {16}{23}{45}
  {16}{24}{35}
  {16}{25}{34}
  {1}{2}{3}{4}{5}{6}

Andrew Howroyd, Table of n, a(n) for n = 1..482 (rows 1..100)

Row sums are A322785. First column is A038041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.

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[Join@@Table[Range[n/d],{d}]],SameQ@@Length/@#&]],{n,10},{d,Divisors[n]}]

T(n,k) = A322794(A002110(n/d)^d), where d = A027750(n,k).

More terms from Alois P. Heinz, Jan 30 2019
Terms a(38) and beyond from Andrew Howroyd, Feb 03 2022
Edited by Peter Munn, Mar 05 2025

A346056 Expansion of e.g.f. Product_{k>=1} B(x^k/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 3, 9, 38, 168, 915, 5225, 34228, 236622, 1805297, 14498751, 125907798, 1146476984, 11129874215, 112934907867, 1209762361679, 13499714095281, 157931096158594, 1918777335806274, 24309294470496502, 318987321135326838, 4346474397776153974

Offset: 0

Seiichi Manyama, Jul 02 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..520

Cf. A000110, A038041, A081066, A209903, A346055, A346058.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k/k!)-1))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, exp(x^k/k!)-1))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!^d))*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!^d))*a(n-k)/(n-k)!));

E.g.f.: exp( Sum_{k>=1} (exp(x^k/k!) - 1) ).
E.g.f.: exp( Sum_{k>=1} A038041(k)*x^k/k! ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!^d)) * a(n-k)/(n-k)! for n > 0.

A346058 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k!)).

Original entry on oeis.org

1, -1, -1, 3, 4, 2, -69, -185, 596, 1482, 22051, -8341, -450570, -1503596, -23829233, 144974757, 150086353, 4859956733, 51013196234, -504522222442, 2572161050316, -58533039862692, 69278113622988, 342581575176372, -25348876024693055, 661312712021911319

Offset: 0

Seiichi Manyama, Jul 02 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..537

Cf. A000587, A038041, A330199, A346056, A346057.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k!)))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k!)))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!^d))*x^k))))

a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!^d))*a(n-k)/(n-k)!));

E.g.f.: exp( Sum_{k>=1} (1 - exp(x^k/k!)) ).
E.g.f.: exp( -Sum_{k>=1} A038041(k)*x^k/k! ).
a(n) = -(n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!^d)) * a(n-k)/(n-k)! for n > 0.

cp's OEIS Frontend

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

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Mathematica

A200472 Triangle T(n,k) is the number of ways to assign n people to k unlabeled groups of equal size.

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A262320 Number of ways to select a subset s from an n-set and then partition s into blocks of equal size.

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Maple

Mathematica

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A299354 Regular triangle where T(n,k) is the number of labeled connected k-uniform hypergraphs spanning n vertices.

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A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.

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Mathematica

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Extensions

A319540 Number of unlabeled 3-uniform hypergraphs spanning n vertices such that every pair of vertices appears together in some block.

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A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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Mathematica

Extensions

A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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