A131632 -id:A131632 - cp's OEIS frontend

A272519 Number of set partitions of [n] into seven blocks with distinct sizes.

2431106898187968000, 8812762505931384000, 67144857188048640000, 416298114565901568000, 3144312274410635328000, 23728992530256389376000, 238675412699786289427200, 3207620559498676985664000, 16207982672116390803648000, 117220515926387332979520000

Offset: 28

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 28..1000

Column k=7 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 7):
seq(a(n), n=28..40);

b[n_, i_, t_] := b[n, i, t] = If[t > i || t*(t + 1)/2 > n || t*(2*i + 1 - t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]*Binomial[n, i]]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 28, 40}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

a(n) = n! * [x^n*y^7] Product_{n>=1} (1+y*x^n/n!).

A272520 Number of set partitions of [n] into eight blocks with distinct sizes.

Original entry on oeis.org

73566121315513295589120000, 302438498741554659644160000, 2585849164240292339957568000, 17681163441201479441398176000, 144230432460463828639480320000, 1126747392322972404668523840000, 11096383744399962905356299840000, 126335461333640259667975104096000

Offset: 36

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 36..1000

Column k=8 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 8):
seq(a(n), n=36..45);

a(n) = n! * [x^n*y^8] Product_{n>=1} (1+y*x^n/n!).

A272521 Number of set partitions of [n] into nine blocks with distinct sizes.

Original entry on oeis.org

65191584694745586153436251091200000, 299881289595829696305806755019520000, 2847357699192726409368266158771200000, 21355182743945448070261996190784000000, 188337379875526275760583438815706112000, 1546608042571126104473205339509472000000

Offset: 45

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 45..1000

Column k=9 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 9):
seq(a(n), n=45..55);

a(n) = n! * [x^n*y^9] Product_{n>=1} (1+y*x^n/n!).

A272522 Number of set partitions of [n] into ten blocks with distinct sizes.

Original entry on oeis.org

1906765806522767212441719098019963758016000000, 9707171378661360354248751771737997313536000000, 101439940907011215701899456014662071926451200000, 829726696136835328433485294068646178065075200000, 7898017947032824439221398664897536276307929600000

Offset: 55

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 55..1000

Column k=10 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 10):
seq(a(n), n=55..65);

a(n) = n! * [x^n*y^10] Product_{n>=1} (1+y*x^n/n!).

A336138 Number of set partitions of the binary indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 5, 2, 4, 5, 12, 1, 2, 2, 5, 2, 5, 4, 13, 2, 4, 5, 13, 5, 13, 13, 43, 1, 2, 2, 5, 2, 5, 5, 13, 2, 5, 4, 14, 5, 13, 14, 42, 2, 4, 5, 13, 5, 14, 13, 43, 5, 13, 14, 45, 14, 44, 44, 160, 1, 2, 2, 5, 2, 5, 5, 14, 2, 5, 5, 14, 4, 13

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 = 3, 7, 11, 15, 23:
  {12}    {123}      {124}      {1234}        {1235}
  {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
          {13}{2}    {12}{4}    {12}{34}      {12}{35}
          {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
                     {1}{2}{4}  {124}{3}      {125}{3}
                                {13}{24}      {13}{25}
                                {134}{2}      {135}{2}
                                {1}{2}{34}    {15}{23}
                                {1}{23}{4}    {1}{2}{35}
                                {1}{24}{3}    {1}{25}{3}
                                {14}{2}{3}    {13}{2}{5}
                                {1}{2}{3}{4}  {15}{2}{3}
                                              {1}{2}{3}{5}

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

The version for twice-partitions is A271619.
The version for partitions of partitions is (also) A271619.
These set partitions are counted by A275780.
The version for factorizations is A321469.
The version for normal multiset partitions is A326519.
The version for equal block-sums is A336137.
Set partitions with distinct block-lengths are A007837.
Set partitions of binary indices are A050315.
Twice-partitions with equal sums are A279787.
Partitions of partitions with equal sums are A305551.
Normal multiset partitions with equal block-lengths are A317583.
Multiset partitions with distinct block-sums are ranked by A326535.
Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.

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]],UnsameQ@@Total/@#&]],{n,0,100}]

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A272519 Number of set partitions of [n] into seven blocks with distinct sizes.

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A272520 Number of set partitions of [n] into eight blocks with distinct sizes.

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A272521 Number of set partitions of [n] into nine blocks with distinct sizes.

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A272522 Number of set partitions of [n] into ten blocks with distinct sizes.

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A336138 Number of set partitions of the binary indices of n with distinct block-sums.

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