A000296 -id:A000296 - cp's OEIS frontend

A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

1, 0, 4, 4, 52, 164, 1364, 7620, 60148, 449252, 3831700, 33811716, 320082228, 3178774564, 33234163668, 363535920196, 4153091085172, 49406896240996, 610777358429204, 7830140410294148, 103914148870277556, 1425254885630973604, 20173671034640405588

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A078944, A194689, A346739, A367890.

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - 4 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-4)^n / k!.
a(0) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(k).

A102287 Total number of even blocks in all partitions of n-set.

Original entry on oeis.org

0, 1, 3, 13, 55, 256, 1274, 6791, 38553, 232171, 1477355, 9898780, 69621864, 512585529, 3940556611, 31560327945, 262805569159, 2271094695388, 20333574916690, 188322882941471, 1801737999086129, 17783472151154007, 180866601699482803, 1893373126840572056

Offset: 1

Vladeta Jovovic, Feb 19 2005

			a(3)=3 because in the 5 (=A000110(3)) partitions 123, (12)/3, (13)/2, 1/(23) and 1/2/3 of {1,2,3} we have 3 blocks of even size (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 1..575

Cf. A005493, A000296.

G:=(cosh(x)-1)*exp(exp(x)-1): Gser:=series(G,x=0,28): seq(n!*coeff(Gser,x^n),n=1..25); # Emeric Deutsch, Jun 22 2005
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0],
       add((p->(p+[0, `if`(i::odd, 0, j)*p[1]]))(
       b(n-i*j, i-1))*multinomial(n, n-i*j, i$j)/j!, j=0..n/i))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 16 2015

Range[0, nn]! CoefficientList[
  D[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], y] /. y -> 1,  x]  (* Geoffrey Critzer, Aug 28 2012 *)

E.g.f: (cosh(x)-1)*exp(exp(x)-1).

More terms from Emeric Deutsch, Jun 22 2005

A178979 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the shortest block has length k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 4, 0, 1, 11, 3, 0, 1, 41, 10, 0, 0, 1, 162, 30, 10, 0, 0, 1, 715, 126, 35, 0, 0, 0, 1, 3425, 623, 56, 35, 0, 0, 0, 1, 17722, 2934, 364, 126, 0, 0, 0, 0, 1, 98253, 15165, 2220, 210, 126, 0, 0, 0, 0, 1, 580317, 86900, 10560, 330, 462, 0, 0, 0, 0, 0, 1

Offset: 1

Geoffrey Critzer, Jan 02 2011

Row sums are Bell numbers A000110.
Column 1 is A000296 (shifted).
From Peter Luschny, Apr 05 2011: (Start)
Sum_{k>1} T(n,k) = A000296(n) count the set partitions with blocks of size > 1.
T(n,1) = A000296(n-1) count the set partitions with blocks of size = 1. Thus for the Bell numbers A000110(n) = Sum_{k>=1} T(n,k) = A000296(n-1) + A000296(n). (End)

			T(4,2) = card ({12|34, 13|24, 14|23}) = 3. - _Peter Luschny_, Apr 05 2011
Triangle begins:
    1;
    1,   1;
    4,   0,  1;
   11,   3,  0,  1;
   41,  10,  0,  0,  1;
  162,  30, 10,  0,  0,  1;
  715, 126, 35,  0,  0,  0,  1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Peter Luschny, Set partitions

Cf. A080510, A145877, A000110, A000296.

g := k-> exp(x)*(1-(GAMMA(k,x)/GAMMA(k))); egf := k-> exp(g(k))-exp(g(k+1));
T := (n,k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, Apr 05 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
       add(b(n-i*j, i+1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> b(n, k) -b(n, k+1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 25 2016

a[k_]:= Exp[x]-Sum[x^i/i!,{i,0,k}]; Transpose[Table[Range[20]! Rest[CoefficientList[Series[Exp[a[k-1]]-Exp[a[k]],{x,0,20}],x]],{k,1,9}]]//Grid

E.g.f. for column k: exp((exp(x) - Sum_{i=0..k-1} x^i/i!)) - exp((exp(x) - Sum_{i=0..k} x^i/i!)).
From Ludovic Schwob, Jan 15 2022: (Start)
T(2n,n) = A001700(n) = C(2n-1,n) for n>0.
T(2n-1,n-1) = A001700(n) = C(2n-1,n) for n>1. (End)

A182930 Triangle read by rows: Number of set partitions of {1,2,..,n} such that |k| is a block and no block |m| with m < k exists, (1 <= n, 1 <= k <= n).

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 5, 3, 2, 1, 15, 10, 7, 5, 4, 52, 37, 27, 20, 15, 11, 203, 151, 114, 87, 67, 52, 41, 877, 674, 523, 409, 322, 255, 203, 162, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 715, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 3425

Offset: 1

Peter Luschny, Apr 08 2011

Mirror image of A106436. - Alois P. Heinz, Jan 29 2019

			T(4,2) = card({2|134, 2|3|14, 2|4|13}) = 3.
[1]     1,
[2]     1,    0,
[3]     2,    1,    1,
[4]     5,    3,    2,    1,
[5]    15,   10,    7,    5,    4,
[6]    52,   37,   27,   20,   15,   11,
     [-1-] [-2-] [-3-] [-4-] [-5-] [-6-]

Alois P. Heinz, Rows n = 1..141, flattened
Peter Luschny, Set partitions

Cf. A000110, A000296, A106436.

T(2n+1,n+1) gives A020556.

T := proc(n, k) option remember; if n = 1 then 1 elif n = k then T(n-1,1) - T(n-1,n-1) else T(n-1,k) + T(n, k+1) fi end:
A182930 := (n,k) -> T(n,k); seq(print(seq(A182930(n,k),k=1..n)),n=1..6);

T[n_, k_] := T[n, k] = Which[n == 1, 1, n == k, T[n-1, 1] - T[n-1, n-1], True, T[n-1, k] + T[n, k+1]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 22 2019 *)

Recursion: The value of T(n,k) is, if n < 0 or k < 0 or k > n undefined, else if n = 1 then 1 else if k = n then T(n-1,1) - T(n-1,n-1); in all other cases T(n,k) = T(n,k+1) + T(n-1,k).

A247491 Number of crossing partitions of {1,2,...,n} that contain no singletons.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 26, 126, 624, 3193, 17119, 96668, 576104, 3621982, 23980620, 166805068, 1215842905, 9263445775, 73599067250, 608471202527, 5224252803246, 46499854580107, 428369819029085, 4078345518655015, 40073659206668916, 405885206895408576, 4232705116291188276

Offset: 0

Peter Luschny, Sep 25 2014

nonn

A partition p of the set {1,2,...,n} whose elements are arranged in their natural order, is crossing if there exist four numbers 1 <= i < k < j < l <= n such that i and j are in the same block, k and l are in the same block, but i,j and k,l belong to two different blocks.

Also the number of crossing partitions of {1,2,...,n} that contain no cyclical adjacencies. e.g., a(5) = 5, [13|24|5, 13|25|4, 14|25|3, 14|2|35, 1|24|35]. - Yuchun Ji, Nov 13 2020

			The crossing partitions of {1,2,3,4,5} that contain no singletons are: [13|245], [14|235], [24,135], [25|134], [35|124].

Indranil Ghosh, Table of n, a(n) for n = 0..163
Peter Luschny, Set partitions

Cf. A016098, A005043, A000110, A000296, A247494.

A247491 := n -> (-1)^n-add((-1)^(n-k)*combinat:-bell(k), k = 0..n-1) - (-1)^n*hypergeom([-n, 1/2], [2], 4); seq(round(evalf(A247491(n), 100)), n=0..27);

Table[Sum[(-1)^(n-k)*Binomial[n,k]*(BellB[k]-CatalanNumber[k]), {k,0,n}], {n, 0, 26}] (* Indranil Ghosh, Mar 04 2017 *)

B(n) = sum(k=0, n, stirling(n,k,2));
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*(B(k)-binomial(2*k,k)/(k+1))); \\ Indranil Ghosh, Mar 04 2017

A247491 = lambda n: sum((-1)^(n-k)*binomial(n,k)*(bell_number(k) - catalan_number(k)) for k in (0..n))
[A247491(n) for n in range(27)]

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(Bell(k)-Catalan(k)).
a(n) = A000296(n) - A005043(n).
a(n) = A016098(n) - A247494(n); i.e., remove the partitions with cyclical adjacencies from the crossing partitions. - Yuchun Ji, Nov 17 2020

A247494 Number of crossing partitions of {1,2,...,n} that contain singletons.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 45, 322, 2086, 13092, 82060, 523116, 3429481, 23279555, 164244262, 1206458632, 9228941572, 73471779239, 608000100209, 5222503739340, 46493341311706, 428345495309624, 4078254436854598, 40073317276815681, 405883920183989049, 4232700263388189325

Offset: 0

Peter Luschny, Oct 02 2014

nonn

A partition p of the set {1,2,...,n} whose elements are arranged in their natural order, is crossing if there exist four numbers 1 <= i < k < j < l <= n such that i and j are in the same block, k and l are in the same block, but i,j and k,l belong to two different blocks.

Also number of crossing partitions of {1,2,...,n} that contain cyclical adjacencies. a(5) = 5, [124|35, 134|25, 135|24, 13|245, 14|235]. - Yuchun Ji, Nov 13 2020

			The crossing partitions of {1,2,3,4,5} that contain singletons are: [1|24|35], [2|14|35], [3|14|25], [4|13|25], [5|13|24].

Indranil Ghosh, Table of n, a(n) for n = 0..170
Peter Luschny, Set partitions

Cf. A016098, A005043, A000110, A000296, A106640, A247491.

A247494 := n -> add((-1)^(n-k+1)*combinat:-bell(k+1), k=0..n-1) + (-1)^n*hypergeom([-n,1/2],[2],4) - binomial(2*n,n)/(n+1):
seq(round(evalf(A247494(n),100)), n=0..25);

Table[Sum[(-1)^(n-k+1)*Binomial[n,k]*(BellB[k]-CatalanNumber[k]),{k,0,n-1}],{n,0,25}] (* Indranil Ghosh, Mar 04 2017 *)

B(n) = sum(k=0, n, stirling(n,k,2));
a(n) = sum(k=0, n-1, (-1)^(n-k+1)*binomial(n,k)*(B(k) - binomial(2*k,k)/(k+1))); \\ Indranil Ghosh, Mar 04 2017

A247494 = lambda n: sum((-1)^(n-k+1)*binomial(n,k)*(bell_number(k)-catalan_number(k)) for k in (0..n-1))
[A247494(n) for n in range(26)]

a(n) = Sum_{k = 0..n-1} (-1)^(n-k+1)*binomial(n,k)*(Bell(k)-Catalan(k)).
a(n) = A016098(n) - A247491(n).
a(n) = A000296(n+1) - A106640(n-1), for n>0 (i.e., remove the non-crossing partitions from the cyclical adjacencies partitions). - Yuchun Ji, Nov 11 2020

A306419 Number of set partitions of {1, ..., n} whose blocks are all singletons and pairs, not including {1, n} or {i, i + 1} for any i.

Original entry on oeis.org

1, 1, 1, 1, 4, 11, 32, 99, 326, 1123, 4064, 15291, 59924, 242945, 1019584, 4409233, 19648674, 89938705, 422744384, 2035739041, 10039057524, 50610247483, 260704414816, 1370387233859, 7346982653702, 40131663286851, 223238920709024, 1263531826402891, 7273434344119460

Offset: 0

Gus Wiseman, Feb 14 2019

Also the number of spanning subgraphs of the complement of an n-cycle, with no overlapping edges.

I.e., for n >= 3, also the number of matchings in the complement of the cycle graph C_n. - Eric W. Weisstein, Sep 02 2025

			The a(1) = 1 through a(5) = 11 set partitions:
  {{1}}  {{1}{2}}  {{1}{2}{3}}  {{13}{24}}      {{1}{24}{35}}
                                {{1}{24}{3}}    {{13}{24}{5}}
                                {{13}{2}{4}}    {{13}{25}{4}}
                                {{1}{2}{3}{4}}  {{14}{2}{35}}
                                                {{14}{25}{3}}
                                                {{1}{2}{35}{4}}
                                                {{1}{24}{3}{5}}
                                                {{1}{25}{3}{4}}
                                                {{13}{2}{4}{5}}
                                                {{14}{2}{3}{5}}
                                                {{1}{2}{3}{4}{5}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000085, A000110, A000296, A001006, A001610, A003436 (no singletons), A034807, A170941 (linear case), A278990 (linear case with no singletons), A306417.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Complement[Subsets[Range[n],{2}],Sort/@Partition[Range[n],2,1,1]],Intersection[#1,#2]!={}&]],{n,0,10}]
(* Second program: *)
CompoundExpression[
  b[n_] := I^(1 - n) 2^((n - 1)/2) HypergeometricU[(1 - n)/2, 3/2, -1/2],
  Join[{1, 1, 1}, Table[Sum[(-1)^k b[n - 2 k] n (n - 1 - k)!/(k! (n - 2 k)!), {k, 0, n/2}], {n, 3, 20}]]
] (* Eric W. Weisstein, Sep 02 2025 *)

\\ here b(n) is A000085(n)
b(n) = {sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}
a(n) = {if(n < 3, n >= 0, sum(k=0, n\2, (-1)^k*b(n-2*k)*n*(n-1-k)!/(k!*(n-2*k)!)))} \\ Andrew Howroyd, Aug 30 2019

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*A034807(n, k)*A000085(n-2*k) for n > 2. - Andrew Howroyd, Aug 30 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 30 2019

A306437 Regular triangle read by rows where T(n,k) is the number of non-crossing set partitions of {1, ..., n} in which all blocks have size k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 5, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 14, 0, 4, 0, 0, 0, 1, 1, 0, 12, 0, 0, 0, 0, 0, 1, 1, 42, 0, 0, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 132, 55, 22, 0, 6, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 429, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Feb 15 2019

			Triangle begins:
  1
  1   1
  1   0   1
  1   2   0   1
  1   0   0   0   1
  1   5   3   0   0   1
  1   0   0   0   0   0   1
  1  14   0   4   0   0   0   1
  1   0  12   0   0   0   0   0   1
  1  42   0   0   5   0   0   0   0   1
  1   0   0   0   0   0   0   0   0   0   1
  1 132  55  22   0   6   0   0   0   0   0   1
Row 6 counts the following non-crossing set partitions (empty columns not shown):
  {{1}{2}{3}{4}{5}{6}}  {{12}{34}{56}}  {{123}{456}}  {{123456}}
                        {{12}{36}{45}}  {{126}{345}}
                        {{14}{23}{56}}  {{156}{234}}
                        {{16}{23}{45}}
                        {{16}{25}{34}}

Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
Wikipedia, Noncrossing partition.

Row sums are A194560. Column k=2 is A126120. Trisection of column k=3 is A001764.

Cf. A000108, A000110, A000296, A001006, A001263, A001610, A016098, A038041, A061095, A125181, A134264.

T:= (n, k)-> `if`(irem(n, k)=0, binomial(n, n/k)/(n-n/k+1), 0):
seq(seq(T(n,k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 16 2019

Table[Table[If[Divisible[n,d],d/n*Binomial[n,n/d-1],0],{d,n}],{n,15}]

If d|n, then T(n, d) = binomial(n, n/d)/(n - n/d + 1); otherwise T(n, k) = 0 [Theorem 1 of Kreweras].

A323949 Number of set partitions of {1, ..., n} with no block containing three distinct cyclically successive vertices.

Original entry on oeis.org

1, 1, 2, 4, 10, 36, 145, 631, 3015, 15563, 86144, 508311, 3180930, 21018999, 146111543, 1065040886, 8117566366, 64531949885, 533880211566, 4587373155544, 40865048111424, 376788283806743, 3590485953393739, 35312436594162173, 357995171351223109, 3736806713651177702

Offset: 0

Gus Wiseman, Feb 10 2019

nonn

Cyclically successive means 1 is a successor of n.

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
         {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                    {{1,3},{2}}    {{1,4},{2,3}}
                    {{1},{2},{3}}  {{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},{2},{3},{4}}

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A000085, A000110, A000126, A000296, A000325, A001610, A001644, A078012, A169985.

Cf. A306351, A306357, A323950, A323951, A323953, A323954, A323955, A323956.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],3,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,8}]

a(12)-a(25) from Alois P. Heinz, Feb 10 2019

A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 4, 10, 14, 15, 11, 36, 46, 51, 52, 41, 145, 184, 196, 202, 203, 162, 631, 806, 855, 869, 876, 877, 715, 3015, 3847, 4059, 4115, 4131, 4139, 4140, 3425, 15563, 19805, 20813, 21056, 21119, 21137, 21146, 21147, 17722, 86144, 109339, 114469

Offset: 1

Gus Wiseman, Feb 10 2019

Cyclically successive means 1 is a successor of n.

			Triangle begins:
    1
    1    2
    1    4    5
    4   10   14   15
   11   36   46   51   52
   41  145  184  196  202  203
  162  631  806  855  869  876  877
  715 3015 3847 4059 4115 4131 4139 4140
Row 4 counts the following partitions:
  {{13}{24}}      {{12}{34}}      {{1}{234}}      {{1234}}
  {{1}{24}{3}}    {{13}{24}}      {{12}{34}}      {{1}{234}}
  {{13}{2}{4}}    {{14}{23}}      {{123}{4}}      {{12}{34}}
  {{1}{2}{3}{4}}  {{1}{2}{34}}    {{124}{3}}      {{123}{4}}
                  {{1}{23}{4}}    {{13}{24}}      {{124}{3}}
                  {{12}{3}{4}}    {{134}{2}}      {{13}{24}}
                  {{1}{24}{3}}    {{14}{23}}      {{134}{2}}
                  {{13}{2}{4}}    {{1}{2}{34}}    {{14}{23}}
                  {{14}{2}{3}}    {{1}{23}{4}}    {{1}{2}{34}}
                  {{1}{2}{3}{4}}  {{12}{3}{4}}    {{1}{23}{4}}
                                  {{1}{24}{3}}    {{12}{3}{4}}
                                  {{13}{2}{4}}    {{1}{24}{3}}
                                  {{14}{2}{3}}    {{13}{2}{4}}
                                  {{1}{2}{3}{4}}  {{14}{2}{3}}
                                                  {{1}{2}{3}{4}}

First column (k = 2) is A000296. Second column (k = 3) is A323949. Rightmost terms are A000110. Second to rightmost terms are A058692.

Cf. A000126, A000325, A001610, A001644, A169985, A306351, A306357, A323950, A323954.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],k,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,7},{k,2,n+1}]

cp's OEIS Frontend

A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

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A102287 Total number of even blocks in all partitions of n-set.

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A178979 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the shortest block has length k (1 <= k <= n).

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A182930 Triangle read by rows: Number of set partitions of {1,2,..,n} such that |k| is a block and no block |m| with m < k exists, (1 <= n, 1 <= k <= n).

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A247491 Number of crossing partitions of {1,2,...,n} that contain no singletons.

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A247494 Number of crossing partitions of {1,2,...,n} that contain singletons.

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A306419 Number of set partitions of {1, ..., n} whose blocks are all singletons and pairs, not including {1, n} or {i, i + 1} for any i.

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