A319375 -id:A319375 - cp's OEIS frontend

A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 7, 7, 1, 21, 25, 13, 1, 66, 101, 71, 21, 1, 258, 366, 396, 166, 31, 1, 1079, 1555, 1877, 1247, 337, 43, 1, 4987, 7099, 9199, 7855, 3305, 617, 57, 1, 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1, 136723, 184033, 253108, 284968, 203278, 79756, 16126, 1666, 91, 1

Offset: 1

Alois P. Heinz, Dec 07 2018

			The 5 set partitions of {1,2,3} are:
  1   |2  |3
  1   |23
  2   |13
  3   |12
  123
so there are 7 elements in the first (smallest) blocks, 7 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
      1;
      3,     1;
      7,     7,     1;
     21,    25,    13,     1;
     66,   101,    71,    21,     1;
    258,   366,   396,   166,    31,    1;
   1079,  1555,  1877,  1247,   337,   43,    1;
   4987,  7099,  9199,  7855,  3305,  617,   57,  1;
  25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Column k=1-10 gives A097147, A332942, A332943, A332944, A332945, A332946, A332947, A332948, A332949, A332950.
Row sums give A070071.
Cf. A319375, A322383.

b:= proc(n, l) option remember; `if`(n=0, add(l[i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j])), j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
      max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
    end:
T:= (n, k)-> b(n, 1, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 02 2020

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)

A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110.

Original entry on oeis.org

0, 1, 4, 15, 60, 260, 1218, 6139, 33120, 190323, 1159750, 7464270, 50563164, 359377681, 2672590508, 20744378175, 167682274352, 1408702786668, 12277382510862, 110822101896083, 1034483164707440, 9972266139291771, 99147746245841106, 1015496134666939958

Offset: 0

Karol A. Penson, Apr 19 2002

nonn

a(n) is the total number of successions among all partitions of {1,2,...,n+1}; a succession is a pair (i,i+1) of consecutive integers lying in a block. For example, a(3)=15 because {1,2,3,4} has 6 partitions with 1 succession - 1/2/34, 1/23/4, 12/3/4, 14/23, 134/2, 124/3, 3 partitions with 2 successions - 1/234, 123/4, 12/34 and 1 partition with 3 successions - 1234. Thus a(3) = 6*1 + 3*2 + 1*3 = 15. - Augustine O. Munagi, Jul 01 2008
a(n) is the number of occurrences of integers in a list of all partitions of the set {1,...,n}. For example, the list 123, 1/23, 2/13, 3/12, 1/2/3 of all partitions of the set {1,2,3} requires 15 occurrences of integers each belonging to that set. [From Michael Hardy (hardy(AT)math.umn.edu), Nov 08 2008]
The bijection between the two foregoing characterizations is as follows: Fix x in {1,2,...,n} and associate x with the succession (x,x+1) which appears in some partitions of {1,2,...,n+1}. Replace x,x+1 by x and partition the n-set {1,2,...,x,x+2,...,n+1}, giving B(n) partitions. Thus the succession (x,x+1) occurs among partitions of {1,2,...,n+1} exactly B(n) times. - Augustine O. Munagi, Jun 02 2010

Alois P. Heinz, Table of n, a(n) for n = 0..574 (terms n=0..200 from Vincenzo Librandi)
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.

Cf. A000110, A052889, A105479, A105480, A105481, A175757.

Row sums of A270236, A270701, A270702, A286416, A319298, A319375.

[n*Bell(n): n in [0..25]]; // Vincenzo Librandi, Mar 15 2014

with(combinat): a:=n->sum(numbcomb (n,0)*bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
with(combinat): a:=n->sum(bell(n), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Apr 25 2007
a:=n->sum(sum(Stirling2(n, k), j=1..n), k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jun 28 2007

a[n_] := n!*Coefficient[Series[x E^(E^x+x-1), {x, 0, n}], x, n]
Table[Sum[BellB[n, 1], {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 16 2009 *)
Table[n*BellB[n], {n, 0, 20}] (* Vaclav Kotesovec, Mar 13 2014 *)

a(n)=local(t); if(n<0,0,t=exp(x+O(x^n)); n!*polcoeff(x*t*exp(t-1),n))

[bell_number(n)*n for n in range(22) ] # Zerinvary Lajos, Mar 14 2009

E.g.f: x*exp(x)*exp(exp(x)-1).
Sum_{k=1..n} n*binomial(n-1, k-1)*Bell(n-k), n >= 2. - Zerinvary Lajos, Nov 22 2006
a(n) ~ n^(n+1) * exp(n/LambertW(n)-1-n) / (sqrt(1+LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{k=1..n} k * A175757(n,k). - Alois P. Heinz, Mar 03 2020
a(n) = Sum_{j=0..n} n * Stirling2(n,j). - Detlef Meya, Apr 11 2024

A097148 Total sum of maximum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 10, 35, 136, 577, 2682, 13435, 72310, 414761, 2524666, 16239115, 109976478, 781672543, 5814797281, 45155050875, 365223239372, 3070422740989, 26780417126048, 241927307839731, 2260138776632752, 21805163768404127, 216970086170175575, 2224040977932468379

Offset: 1

Vladeta Jovovic, Jul 27 2004

Let M be the infinite lower triangular matrix given by A080510 and v the column vector [1,2,3,...] then M*v=A097148 (this sequence, as column vector). - Gary W. Adamson, Feb 24 2011

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A028417, A028418, A046746, A006128, A097145-A097147.
Cf. A080510.
Column k=1 of A319375.

b:= proc(n, m) option remember; `if`(n=0, m, add(
      b(n-j, max(j, m))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Mar 02 2020, revised May 08 2024

Drop[ Range[0, 22]! CoefficientList[ Series[ Sum[E^(E^x - 1) - E^Sum[x^j/j!, {j, 1, k}], {k, 0, 22}], {x, 0, 22}], x], 1] (* Robert G. Wilson v, Aug 05 2004 *)

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

More terms from Robert G. Wilson v, Aug 05 2004

A333059 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 4, 17, 76, 357, 1737, 8997, 49420, 289253, 1793221, 11727861, 80576965, 579781009, 4356513727, 34118896917, 277963716808, 2351740613433, 20630800971825, 187374815249205, 1759353644746663, 17055176943817785, 170477858336708555, 1754992340756441973

Offset: 2

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..576
Wikipedia, Partition of a set

Column k=2 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 2)[2]:
seq(a(n), n=2..24);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
     Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
     multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
     b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 2][[2]];
a /@ Range[2, 24] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)

A333060 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 7, 36, 186, 1023, 5867, 34744, 211888, 1343046, 8896185, 61801182, 449917898, 3425580850, 27183592435, 224196765392, 1917038645772, 16963064269986, 155112925687673, 1464150720422785, 14253033440621462, 142967758696293317, 1476398153663677539

Offset: 3

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..576
Wikipedia, Partition of a set

Column k=3 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 3)[2]:
seq(a(n), n=3..24);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
     Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
     multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
     b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 3][[2]];
a /@ Range[3, 24] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)

A333061 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 11, 81, 512, 3151, 20071, 133853, 924320, 6551293, 47529561, 354259153, 2725545695, 21741995463, 180198265559, 1551865576121, 13865702570254, 128238585735637, 1224733005946425, 12053244176971825, 122035994844818345, 1269623551116437475, 13561114665253219451

Offset: 4

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..576
Wikipedia, Partition of a set

Column k=4 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 4)[2]:
seq(a(n), n=4..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
     Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
     multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
     b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 4][[2]];
a /@ Range[4, 25] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)

A333062 Number of entries in the fifth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 16, 162, 1345, 10096, 72973, 531015, 3984762, 30987321, 248303940, 2036778980, 17044330217, 145588640408, 1272940217747, 11434350878640, 105849240653792, 1011701166471075, 9987958951272492, 101765834737586068, 1068365602976497915, 11534318293877771406

Offset: 5

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..576
Wikipedia, Partition of a set

Column k=5 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 5)[2]:
seq(a(n), n=5..25);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[
     Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
     multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
     b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 5][[2]];
a /@ Range[5, 25] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)

A333063 Number of entries in the sixth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 22, 295, 3145, 29503, 256565, 2144517, 17743090, 148599335, 1276302900, 11282648837, 102385155537, 949462521827, 8967298797097, 86161326118467, 843025151446964, 8418457337349711, 86033922399717781, 902026616406147607, 9718711403938257151

Offset: 6

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..576
Wikipedia, Partition of a set

Column k=6 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 6)[2]:
seq(a(n), n=6..25);

A333064 Number of entries in the seventh blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 29, 499, 6676, 77078, 810470, 8016373, 76334142, 713507667, 6658565009, 62882380589, 606149817728, 5983648334738, 60440402586898, 622934996801505, 6532386995235676, 69575530733726891, 752420279343383619, 8269751757387345271, 92538014365261646366

Offset: 7

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..576
Wikipedia, Partition of a set

Column k=7 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 7)[2]:
seq(a(n), n=7..26);

A333065 Number of entries in the eighth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

Original entry on oeis.org

1, 37, 796, 13091, 183074, 2300949, 26869727, 298009584, 3190196105, 33408754043, 346447486658, 3596858467639, 37729664292626, 402420037089997, 4378064125023471, 48598561597491856, 549602546782048959, 6318831653757436761, 73723789208978689966

Offset: 8

Alois P. Heinz, Mar 06 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..576
Wikipedia, Partition of a set

Column k=8 of A319375.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 8)[2]:
seq(a(n), n=8..26);

cp's OEIS Frontend

A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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Maple

Mathematica

A070071 a(n) = n*B(n), where B(n) are the Bell numbers, A000110.

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A097148 Total sum of maximum block sizes in all partitions of n-set.

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A333059 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

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Maple

Mathematica

A333060 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

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Maple

Mathematica

A333061 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

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Maple

Mathematica

A333062 Number of entries in the fifth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

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Maple

Mathematica

A333063 Number of entries in the sixth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

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