A007837 -id:A007837 - cp's OEIS frontend

A358830 Number of twice-partitions of n into partitions with all different lengths.

1, 1, 2, 4, 9, 15, 31, 53, 105, 178, 330, 555, 1024, 1693, 2991, 5014, 8651, 14242, 24477, 39864, 67078, 109499, 181311, 292764, 483775, 774414, 1260016, 2016427, 3254327, 5162407, 8285796, 13074804, 20812682, 32733603, 51717463, 80904644, 127305773, 198134675, 309677802

Offset: 0

Gus Wiseman, Dec 03 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)
                      (11)(2)   (2111)
                      (2)(11)   (11111)
                      (21)(1)   (21)(2)
                      (111)(1)  (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)

The version for set partitions is A007837.
For sums instead of lengths we have A271619.
For constant instead of distinct lengths we have A306319.
The case of distinct sums also is A358832.
The version for multiset partitions of integer partitions is A358836.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A273873 counts strict trees.
Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A336342, A358334, A358831.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],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,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

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

A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 15, 1, 21, 60, 1, 63, 105, 1, 92, 448, 1, 255, 2016, 1, 385, 4980, 12600, 1, 1023, 15675, 27720, 1, 1585, 61644, 138600, 1, 4095, 155155, 643500, 1, 6475, 482573, 4408404, 1, 16383, 1733550, 12687675, 37837800, 1, 26332, 4549808, 60780720

Offset: 1

Vladeta Jovovic, Sep 04 2007

Row sums = A007837.
Sum k! * T(n,k) = A032011.
Sum k * T(n,k) = A131623. - Geoffrey Critzer, Aug 30 2012.
T(n,k) is also the number of words w of length n over a k-ary alphabet {a1,a2,...,ak} with #(w,a1) > #(w,a2) > ... > #(w,ak) > 0, where #(w,x) counts the letters x in word w.  T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa. - Alois P. Heinz, Jun 21 2013

			Triangle T(n,k)begins:
  1;
  1;
  1,     3;
  1,     4;
  1,    15;
  1,    21,      60;
  1,    63,     105;
  1,    92,     448;
  1,   255,    2016;
  1,   385,    4980,    12600;
  1,  1023,   15675,    27720;
  1,  1585,   61644,   138600;
  1,  4095,  155155,   643500;
  1,  6475,  482573,  4408404;
  1, 16383, 1733550, 12687675, 37837800;
  ...

Alois P. Heinz, Rows n = 1..500, flattened

Columns k=1-10 give: A000012, A272514, A272515, A272516, A272517, A272518, A272519, A272520, A272521, A272522.

Cf. A000217, A003056, A007837, A022915, A032011, A131623, A226873, A226874, A327803.

b:= proc(n, i, t, v) option remember; `if`(t=1, 1/(n+v)!,
      add(b(n-j, j, t-1, v+1)/(j+v)!, j=i..n/t))
    end:
T:= (n, k)->`if`(k*(k+1)/2>n, 0, n!*b(n-k*(k+1)/2, 0, k, 1)):
seq(seq(T(n, k), k=1..floor(sqrt(2+2*n)-1/2)), n=1..20);
# Alois P. Heinz, Jun 21 2013
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Sep 27 2019

nn=10;p=Product[1+y x^i/i!,{i,1,nn}];Range[0,nn]! CoefficientList[ Series[p,{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Aug 30 2012 *)

E.g.f.: Product_{n>=1} (1+y*x^n/n!).

T(A000217(n),n) = A022915(n). - Alois P. Heinz, Jul 03 2018

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

A032011 Partition n labeled elements into sets of different sizes and order the sets.

Original entry on oeis.org

1, 1, 1, 7, 9, 31, 403, 757, 2873, 12607, 333051, 761377, 3699435, 16383121, 108710085, 4855474267, 13594184793, 76375572751, 388660153867, 2504206435681, 20148774553859, 1556349601444477, 5050276538344665, 33326552998257031, 186169293932977115, 1305062351972825281, 9600936552132048553, 106019265737746665727, 12708226588208611056333, 47376365554715905155127

Offset: 0

Christian G. Bower, Apr 01 1998

nonn

From Alois P. Heinz, Sep 02 2015: (Start)
Also the number of matrices with n rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct.  Equivalently, the number of compositions of n into distinct parts where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once.
a(3) = 7:
[1]   [1 0]  [0 1]  [1 0]  [0 1]  [0 1]  [1 0]
[1]   [1 0]  [0 1]  [0 1]  [1 0]  [1 0]  [0 1]
[1]   [0 1]  [1 0]  [1 0]  [0 1]  [1 0]  [0 1].
3abc, 2ab1c, 1c2ab, 2ac1b, 1b2ac, 2bc1a, 1a2bc.  (End)

Alois P. Heinz, Table of n, a(n) for n = 0..670
C. G. Bower, Transforms (2)

Cf. A000670, A007837, A032020, A114902, A120774, A131632.

Main diagonal of A261836 and A261959.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 02 2015

f[list_]:=Apply[Multinomial,list]*Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 1, 30}]
b[n_, i_, p_] := b[n, i, p] = If[i*(i+1)/2n, 0, b[n-i, i-1, p+1]*Binomial[n, i]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

seq(n)=[subst(serlaplace(y^0*p),y,1) | p <- Vec(serlaplace(prod(k=1, n, 1 + x^k*y/k! + O(x*x^n))))] \\ Andrew Howroyd, Sep 13 2018

"AGJ" (ordered, elements, labeled) transform of 1, 1, 1, 1, ...

a(n) = Sum_{k>=0} k! * A131632(n,k). - Alois P. Heinz, Sep 09 2015

a(0)=1 prepended by Alois P. Heinz, Sep 02 2015

A178682 The number of functions f:{1,2,...,n}->{1,2,...,n} such that the number of elements that are mapped to m is divisible by m.

Original entry on oeis.org

1, 1, 2, 5, 13, 42, 150, 576, 2266, 9966, 47466, 237019, 1224703, 6429152, 35842344, 212946552, 1325810173, 8488092454, 55276544436, 362961569008, 2465240278980, 17538501945077, 130454679958312, 1002493810175093, 7838007702606372, 61789072382062638

Offset: 0

Geoffrey Critzer, Dec 25 2010

a(n) is also the number of partitions of n where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition. a(3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc. There is a simple bijection between the marked partitions and the functions f. - Alois P. Heinz, Aug 30 2015

			a(3) = 5 because there are 5 such functions: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (3,3,3).
G.f. = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 42*x^5 + 150*x^6 + 576*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..680

Cf. A000110, A005651, A007837, A261774.
Main diagonal of A326500, A326616, A326617.
Row sums of A364285, A364310.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&*[(&+[x^(k*j)/Factorial(k*j): k in [0..m]]): j in [1..m]]) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jan 26 2019

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 30 2015

Range[0,20]! CoefficientList[Series[Product[Sum[x^(j i)/(j i)!,{i,0,20}],{j,1,20}],{x,0,20}],x]

m=30; my(x='x+O('x^m)); Vec(serlaplace(prod(j=1, m, sum(k=0,m, x^(k*j)/(k*j)!)))) \\ G. C. Greubel, Jan 26 2019

m = 30; T = taylor(product(sum(x^(k*j)/factorial(k*j) for k in (0..m)) for j in (1..m)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jan 26 2019

E.g.f.: Product_{j>=1} Sum_{i>=0} x^(j*i)/(j*i)!.

a(21)-a(25) from Alois P. Heinz, Aug 30 2015

A271423 Number T(n,k) of set partitions of [n] with maximal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 16, 25, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 169, 406, 245, 35, 21, 0, 1, 0, 541, 2093, 1036, 385, 56, 28, 0, 1, 0, 2272, 10935, 4984, 2331, 504, 84, 36, 0, 1, 0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1

Offset: 0

Alois P. Heinz, Apr 07 2016

At least one block length occurs exactly k times, and all block lengths occur at most k times.

			T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
T(4,2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,     1;
  0,     5,     9,     0,     1;
  0,    16,    25,    10,     0,    1;
  0,    82,    70,    35,    15,    0,   1;
  0,   169,   406,   245,    35,   21,   0,   1;
  0,   541,  2093,  1036,   385,   56,  28,   0,  1;
  0,  2272, 10935,  4984,  2331,  504,  84,  36,  0, 1;
  0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A007837 (for n>0), A271731, A271732, A271733, A271734, A271735, A271736, A271737, A271738, A271739.
Row sums give A000110.
Main diagonal gives A000012.
T(2n,n) gives A271425.
Cf. A271424.

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]] * b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i]}]]]; T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

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}]

A326533 MM-numbers of multiset partitions where each part has a different length.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119, 122

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

These are numbers where each prime index has a different Omega (number of prime factors counted with multiplicity). 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 average 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}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  31: {{5}}
  34: {{},{4}}
  35: {{2},{1,1}}

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

A subsequence of A005117.

Cf. A007837, A038041, A112798, A302242, A320324, A326026, A326514, A326517, A326534, A326535, A326536, A326537.

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

A336137 Number of set partitions of the binary indices of n with equal block-sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

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 a(n) set partitions for n = 7, 59, 119, 367, 127:
  {123}    {12456}      {123567}      {1234679}    {1234567}
  {12}{3}  {126}{45}    {1236}{57}    {12346}{79}  {1247}{356}
           {15}{24}{6}  {156}{237}    {1249}{367}  {1256}{347}
                        {17}{26}{35}  {1267}{349}  {1346}{257}
                                      {169}{2347}  {167}{2345}
                                                   {16}{25}{34}{7}
The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
  {{2,7},{3,6},{4,5},{9}}
  {{2,4,6},{3,9},{5,7}}
  {{2,7,9},{3,4,5,6}}
  {{2,3,4,9},{5,6,7}}
  {{2,3,6,7},{4,5,9}}
  {{2,4,5,7},{3,6,9}}
  {{2,3,4,5,6,7,9}}
so a(382) = 7.

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

These set partitions are counted by A035470.
The version for twice-partitions is A279787.
The version for partitions of partitions is A305551.
The version for factorizations is A321455.
The version for normal multiset partitions is A326518.
The version for distinct block-sums is A336138.
Set partitions of binary indices are A050315.
Normal multiset partitions with equal lengths are A317583.
Normal multiset partitions with equal averages are A326520.
Multiset partitions with equal block-sums are ranked by A326534.
Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]

cp's OEIS Frontend

A358830 Number of twice-partitions of n into partitions with all different lengths.

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

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A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

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

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A032011 Partition n labeled elements into sets of different sizes and order the sets.

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A178682 The number of functions f:{1,2,...,n}->{1,2,...,n} such that the number of elements that are mapped to m is divisible by m.

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A271423 Number T(n,k) of set partitions of [n] with maximal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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