A250104 -id:A250104 - cp's OEIS frontend

A250105 Column 1 of triangle in A250104 (or A124323).

0, 0, 3, 4, 20, 66, 287, 1296, 6435, 34250, 194942, 1179036, 7544121, 50865920, 360167355, 2670210640, 20673196460, 166753291806, 1398415162703, 12169520162440, 109709590135635, 1022997624845614, 9852508254721222, 97880299543896216, 1001841501018883425

Offset: 1

N. J. A. Sloane, Nov 16 2014

nonn

T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A250104, A124323.

a[n_] := n*(Sum[(-1)^(j-1)*BellB[n-j-1], {j, 1, n-1}]-(-1)^n); a[1] = 0; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Dec 09 2014 *)

A250107 Column 3 of triangle in A250104 (or A124323).

Original entry on oeis.org

1, 0, 10, 20, 140, 616, 3444, 19440, 117975, 753500, 5068492, 35764092, 264044235, 2034636800, 16327586760, 136180742640, 1178372198220, 10561041814380, 97889061389210, 937053052507880, 9252175434771885, 94115781485796488, 985250825472122200

Offset: 3

N. J. A. Sloane, Nov 16 2014

nonn

T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A250104, A124323.

A250107 := proc(n)
    A124323(n,3) ;
end proc:
seq(A250107(n),n=3..50) ; # R. J. Mathar, Jan 22 2015

t[0, 0] = 0; t[1, 0] = 1; t[1, 1] = 0;
t[n_, k_] := Binomial[n, k] ((-1)^(n-k) + Sum[(-1)^(j-1) BellB[n-k-j], {j, 1, n-k}]);
Table[t[n, 3], {n, 3, 25}] (* Jean-François Alcover, Mar 30 2020 *)

A250106 Column 2 of triangle in A250104 (or A124323).

Original entry on oeis.org

1, 0, 6, 10, 60, 231, 1148, 5832, 32175, 188375, 1169652, 7663734, 52808847, 381494400, 2881338840, 22696790440, 186058768140, 1584156272157, 13984151627030, 127779961705620, 1206805491491985, 11764472685724561, 118230099056654664

Offset: 2

N. J. A. Sloane, Nov 16 2014

nonn

T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A250104, A124323.

A250106 := proc(n)
    A124323(n,2) ;
end proc:
seq(A250106(n),n=2..50) ; # R. J. Mathar, Jan 22 2015

t[n_, k_] := Binomial[n, k] ((-1)^(n-k) + Sum[(-1)^(j-1) BellB[n-k-j], {j, 1, n-k}]);
Table[t[n, 2], {n, 2, 24}] (* Jean-François Alcover, Apr 03 2020 *)

A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 6, 0, 1, 11, 20, 10, 10, 0, 1, 41, 66, 60, 20, 15, 0, 1, 162, 287, 231, 140, 35, 21, 0, 1, 715, 1296, 1148, 616, 280, 56, 28, 0, 1, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 0, 1, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 0, 1

Offset: 0

Emeric Deutsch, Oct 28 2006

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).
Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009
Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014
Also the number of set partitions of {1, ..., n} with k cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). Unlike A250104, we count {{1}} as having 1 cyclical adjacency. - Gus Wiseman, Feb 13 2019

			T(4,2)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34 (if we take {1,2,3,4} as our 4-set).
Triangle starts:
     1
     0    1
     1    0    1
     1    3    0    1
     4    4    6    0    1
    11   20   10   10    0    1
    41   66   60   20   15    0    1
   162  287  231  140   35   21    0    1
   715 1296 1148  616  280   56   28    0    1
  3425 6435 5832 3444 1386  504   84   36    0    1
From _Gus Wiseman_, Feb 13 2019: (Start)
Row n = 5 counts the following set partitions by number of singletons:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{12}{34}}  {{123}{4}}  {{1}{23}{4}}
  {{13}{24}}  {{124}{3}}  {{12}{3}{4}}
  {{14}{23}}  {{134}{2}}  {{1}{24}{3}}
                          {{13}{2}{4}}
                          {{14}{2}{3}}
... and the following set partitions by number of cyclical adjacencies:
  {{13}{24}}      {{1}{2}{34}}  {{1}{234}}  {{1234}}
  {{1}{24}{3}}    {{1}{23}{4}}  {{12}{34}}
  {{13}{2}{4}}    {{12}{3}{4}}  {{123}{4}}
  {{1}{2}{3}{4}}  {{14}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
(End)
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is
0, 1,
1, 0, 1,
1, 2, 0, 1,
1, 3, 3, 0, 1,
1, 4, 6, 4, 0, 1,
1, 5, 10, 10, 5, 0, 1,
1, 6, 15, 20, 15, 6, 0, 1,
1, 7, 21, 35, 35, 21, 7, 0, 1,
1, 8, 28, 56, 70, 56, 28, 8, 0, 1 (End)

Alois P. Heinz, Rows n = 0..140, flattened
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A000110, A052889, A124324.
A250104 is an essentially identical triangle, differing only in row 1.
For columns see A000296, A250105, A250106, A250107.
Cf. A000126, A001610, A032032, A052841, A066982, A080107, A169985, A187784, A324011, A324014, A324015.

G:=exp(exp(z)-1+(t-1)*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form
# Program from R. J. Mathar, Jan 22 2015:
A124323 := proc(n,k)
    binomial(n,k)*A000296(n-k) ;
end proc:

Flatten[CoefficientList[Range[0,10]! CoefficientList[Series[Exp[x y] Exp[Exp[x] - x - 1], {x, 0,10}], x], y]] (* Geoffrey Critzer, Nov 24 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Count[#,{}]==k&]],{n,0,9},{k,0,n}] (* _Gus Wiseman, Feb 13 2019 *)

T(n,k) = binomial(n,k)*[(-1)^(n-k)+sum((-1)^(j-1)*B(n-k-j), j=1..n-k)], where B(q) are the Bell numbers (A000110).
E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)*z).
G.f.: 1/(1-xy-x^2/(1-xy-x-2x^2/(1-xy-2x-3x^2/(1-xy-3x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009

cp's OEIS Frontend

A250105 Column 1 of triangle in A250104 (or A124323).

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A250107 Column 3 of triangle in A250104 (or A124323).

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A250106 Column 2 of triangle in A250104 (or A124323).

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A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

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