A319298 -id:A319298 - cp's OEIS frontend

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

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

A319375 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 3, 1, 10, 4, 1, 35, 17, 7, 1, 136, 76, 36, 11, 1, 577, 357, 186, 81, 16, 1, 2682, 1737, 1023, 512, 162, 22, 1, 13435, 8997, 5867, 3151, 1345, 295, 29, 1, 72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1, 414761, 289253, 211888, 133853, 72973, 29503, 6676, 796, 46, 1

Offset: 1

Alois P. Heinz, Dec 07 2018

			The 5 set partitions of {1,2,3} are:
  1   |2  |3
  12  |3
  13  |2
  23  |1
  123
so there are 10 elements in the first (largest) blocks, 4 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
      1;
      3,     1;
     10,     4,     1;
     35,    17,     7,     1;
    136,    76,    36,    11,     1;
    577,   357,   186,    81,    16,    1;
   2682,  1737,  1023,   512,   162,   22,   1;
  13435,  8997,  5867,  3151,  1345,  295,  29,  1;
  72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1;
  ...

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

Columns k=1-10 give: A097148, A333059, A333060, A333061, A333062, A333063, A333064, A333065, A333066, A333067.
Row sums give A070071.
Cf. A319298, A322384.

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<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:
T:= (n, k)-> b(n$2, 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, {}]];
Array[T, 12] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)

A097147 Total sum of minimum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 7, 21, 66, 258, 1079, 4987, 25195, 136723, 789438, 4863268, 31693715, 217331845, 1564583770, 11795630861, 92833623206, 760811482322, 6479991883525, 57256139503047, 523919025038279, 4956976879724565, 48424420955966635, 487810283307069696

Offset: 1

Vladeta Jovovic, Jul 27 2004

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

Cf. A028417, A028418, A046746, A006128, A097145, A097146, A097148.

Column k=1 of A319298.

g:= proc(n, i, p) option remember; `if`(n=0, (i+1)*p!,
      `if`(i<1, 0, add(g(n-i*j, i-1, p+j*i)/j!/i!^j, j=0..n/i)))
    end:
a:= n-> g(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 06 2015

Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i!,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1]  (* Geoffrey Critzer, Jan 10 2013 *)
g[n_, i_, p_] := g[n, i, p] = If[n == 0, (i+1)*p!, If[i<1, 0,
     Sum[g[n-i*j, i-1, p+j*i]/j!/i!^j, {j, 0, n/i}]]];
a[n_] := g[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Aug 24 2021, after Alois P. Heinz *)

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

More terms from Max Alekseyev, Apr 29 2010

A332942 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 7, 25, 101, 366, 1555, 7099, 34627, 184033, 1059972, 6425992, 41266681, 280938451, 2009636335, 15025372685, 117386912433, 956458929950, 8104399834719, 71244441818927, 648761935841876, 6110827367541999, 59454153443971106, 596654820386392152

Offset: 2

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=2 of A319298.

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:
a:= n-> b(n, 1, 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 > n, 0 {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 2][[2]];
a /@ Range[2, 24] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)

A332943 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 13, 71, 396, 1877, 9199, 47371, 253108, 1420475, 8598976, 55100124, 369764734, 2614650820, 19473708445, 151503397725, 1226996194292, 10339319950504, 90530421514787, 821670728202320, 7714779905351852, 74815825933883534, 748526174347790448, 7717807072187843156

Offset: 3

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=3 of A319298.

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:
a:= n-> b(n, 1, 3)[2]:
seq(a(n), n=3..25);

A332944 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 21, 166, 1247, 7855, 47245, 284968, 1741235, 10782872, 69537976, 471717130, 3336898255, 24584784957, 189704257763, 1530649634720, 12849873769593, 111945035887787, 1011184665775833, 9458811859041042, 91480934118104305, 913112230809837136, 9391472034599656856

Offset: 4

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=4 of A319298.

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:
a:= n-> b(n, 1, 4)[2]:
seq(a(n), n=4..25);

A332945 Number of entries in the fifth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 31, 337, 3305, 27085, 203278, 1470470, 10525307, 74735025, 534251602, 3917281240, 29620189877, 230717097457, 1858373700800, 15544789470865, 135021858204377, 1215283222207344, 11317462847981403, 108929278516177839, 1082642589072326140, 11099778977689173356

Offset: 5

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=5 of A319298.

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:
a:= n-> b(n, 1, 5)[2]:
seq(a(n), n=5..25);

A332946 Number of entries in the sixth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 43, 617, 7681, 79756, 741665, 6467891, 54658254, 451897330, 3685879069, 30091146181, 248749105815, 2091117462980, 17933165800591, 157654535847037, 1426401197217090, 13303368764700743, 127934361462621048, 1268098183967052868, 12948542410221048226

Offset: 6

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=6 of A319298.

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:
a:= n-> b(n, 1, 6)[2]:
seq(a(n), n=6..25);

A332947 Number of entries in the seventh blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 57, 1045, 16126, 207131, 2351933, 24592218, 244969609, 2361155669, 22194089013, 205472269667, 1894376413748, 17523281149447, 163348288354057, 1541060427442896, 14781379209476531, 144692817107094283, 1449416720608086968, 14882752251954912426

Offset: 7

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=7 of A319298.

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:
a:= n-> b(n, 1, 7)[2]:
seq(a(n), n=7..25);

A332948 Number of entries in the eighth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Original entry on oeis.org

1, 73, 1666, 31241, 486377, 6638568, 82413787, 961757669, 10756109317, 116409307679, 1228840258412, 12766418662681, 131564790987337, 1352403410824806, 13925656629847559, 144227711099572501, 1508401366805906552, 15986784485974156076, 172225656206792090516

Offset: 8

Alois P. Heinz, Mar 03 2020

nonn

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

Column k=8 of A319298.

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:
a:= n-> b(n, 1, 8)[2]:
seq(a(n), n=8..26);

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A319375 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A097147 Total sum of minimum block sizes in all partitions of n-set.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A332942 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A332943 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A332944 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A332945 Number of entries in the fifth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A332946 Number of entries in the sixth blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A332947 Number of entries in the seventh blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

Views

Author