A124323 -id:A124323 - cp's OEIS frontend

A124324 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)).

1, 1, 1, 1, 1, 4, 1, 11, 3, 1, 26, 25, 1, 57, 130, 15, 1, 120, 546, 210, 1, 247, 2037, 1750, 105, 1, 502, 7071, 11368, 2205, 1, 1013, 23436, 63805, 26775, 945, 1, 2036, 75328, 325930, 247555, 27720, 1, 4083, 237127, 1561516, 1939630, 460845, 10395, 1, 8178

Offset: 0

Emeric Deutsch, Oct 28 2006

Row sums are the Bell numbers (A000110).
It appears that the triangles in this sequence and A112493 have identical columns, except for shifts. - Jörgen Backelin, Jun 20 2022
Equivalent to Jörgen Backelin's observation, the rows of A112493 may be read off as the diagonals of this entry. - Tom Copeland, Sep 24 2022

			T(4,2) = 3 because we have 12|34, 13|24 and 14|23 (if we take {1,2,3,4} as our 4-set).
Triangle starts:
  1;
  1;
  1,    1;
  1,    4;
  1,   11,     3;
  1,   26,    25;
  1,   57,   130,    15;
  1,  120,   546,   210;
  1,  247,  2037,  1750,   105;
  1,  502,  7071, 11368,  2205;
  1, 1013, 23436, 63805, 26775, 945;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Per Alexandersson and Olivia Nabawanda, Peaks are preserved under run-sorting, arXiv:2104.04220 [math.CO], 2021.
Fufa Beyene and Roberto Mantaci, Merging-Free Partitions and Run-Sorted Permutations, arXiv:2101.07081 [math.CO], 2021.
Tom Copeland, Appell-Bell polynomials: Linking the associated Bell polynomials and the associated reduced inverse refined Eulerian polynomials, 2022.
Tom Copeland, The reduced inverse refined Eulerian polynomials and associated arrays, 2022.
Robin Houston, Adam P. Goucher, and Nathaniel Johnston, A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks, arXiv:2301.06586 [math.CO], 2023.
O. Nabawanda, F. Rakotondrajao, and A. S. Bamunoba, Run Distribution Over Flattened Partitions, arXiv:2007.03821 [math.CO], 2020.

Columns k=0-10 give: A000012, A000295, A112495, A112496, A112497, A290034, A290035, A290036, A290037, A290038, A290039.

Cf. A000110, A001147, A048993, A124323, A124325, A136394, A112493.

G:=exp(t*exp(z)-t+(1-t)*z): Gser:=simplify(series(G,z=0,36)): for n from 0 to 33 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 13 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      `if`(i>1, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015, Jul 15 2017

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] :=  b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1]*If[i>1, x^j, 1], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

E.g.f.: G(t,z) = exp(t*exp(z) - t + (1-t)*z).
T(n,1) = A000295(n) (the Eulerian numbers).
Sum_{k=0..floor(n/2)} k*T(n,k) = A124325(n).
T(2n,n) = A001147(n). - Alois P. Heinz, Apr 06 2018

A324011 Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 5, 14, 66, 307, 1554, 8415, 48530, 296582, 1913561, 12988776, 92467629, 688528288, 5349409512, 43270425827, 363680219762, 3170394634443, 28619600156344, 267129951788160, 2574517930001445, 25587989366964056, 261961602231869825

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

These set partitions are fixed points under Callan's bijection phi on set partitions.

			The a(4) = 1, a(6) = 5, and a(7) = 14 set partitions:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296 (singletons allowed, or adjacencies allowed), A001610, A124323, A169985, A261139, A324012, A324014, A324015.

Table[Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

a(11)-a(26) from Alois P. Heinz, Feb 12 2019

A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 17, 9, 0, 1, 169, 68, 18, 0, 1, 2079, 845, 170, 30, 0, 1, 31261, 12474, 2535, 340, 45, 0, 1, 554483, 218827, 43659, 5915, 595, 63, 0, 1, 11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1, 262517615, 102030777, 19961388, 2625924, 261954, 21294, 1428, 108, 0, 1

Offset: 0

Alois P. Heinz, Dec 19 2021

			T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
Triangle T(n,k) begins:
         1;
         0,       1;
         3,       0,      1;
        17,       9,      0,      1;
       169,      68,     18,      0,     1;
      2079,     845,    170,     30,     0,   1;
     31261,   12474,   2535,    340,    45,   0,  1;
    554483,  218827,  43659,   5915,   595,  63,  0, 1;
  11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: |A069856|, A348590.
Row sums give A000312.
T(n+1,n-1) gives A045943.
Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
      b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
# second Maple program:
A350212 := (n,k)-> add((-1)^(j-k)*binomial(j,k)*binomial(n,j)*(n-j)^(n-j), j=0..n):
seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # Mélika Tebni, Nov 24 2022

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*
     b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A055897(n).
Sum_{k=1..n} T(n,k) = A350134(n).
From Mélika Tebni, Nov 24 2022: (Start)
T(n,k) = binomial(n, k)*|A069856(n-k)|.
E.g.f. column k: exp(-x)*x^k / ((1 + LambertW(-x))*k!).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)

A184174 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks of size 2, i.e., blocks of the form (i,i+1) (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 10, 4, 1, 35, 14, 3, 139, 54, 9, 1, 611, 224, 38, 4, 2925, 1027, 171, 16, 1, 15128, 5112, 822, 80, 5, 83903, 27352, 4279, 415, 25, 1, 495929, 156392, 23826, 2272, 145, 6, 3108129, 950285, 141039, 13252, 855, 36, 1, 20565721, 6107540, 883982, 81692, 5257, 238, 7

Offset: 0

Emeric Deutsch, Feb 09 2011

Row n contains 1 + floor(n/2) entries.
Sum of entries in row n = A000110(n) (the Bell numbers).
T(n,0) = A184175(n).
Sum_{k>=0} k*T(n,k) = A052889(n-1).

			T(4,1)=4 because we have 12-3-4, 1-23-4, 1-2-34, 14-23. T(4,2)=1 because we have 12-34.
Triangle starts:
1;
1;
1, 1;
3, 2;
10, 4, 1;
35, 14, 3;
139, 54, 9, 1;
611, 224, 38, 4;
2925, 1027, 171, 16, 1;
15128, 5112, 822, 80, 5;
83903, 27352, 4279, 415, 25, 1;
495929, 156392, 23826, 2272, 145, 6;
3108129, 950285, 141039, 13252, 855, 36, 1; ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000110, A184175, A052889, A124323, A184176, A184177.

with(combinat): q := 2: a := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*binomial(n+j-j*q, j)*bell(n-j*q), j = k .. floor(n/q)) end proc: for n from 0 to 13 do seq(a(n, k), k = 0 .. floor(n/q)) end do; # yields sequence in triangular form

T[n_, k_] := Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[n-j, j]*BellB[n-2j], {j, k, Floor[n/2]}]; Table[T[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Feb 21 2017 *)

{T(n,k) = my(A = sum(m=0,n, x^m/prod(k=0,m,1 - k*x + (1-y)*x^2 +x*O(x^n)))); polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,12,for(k=0,n\2,print1(T(n,k),", "));print("")) \\ Paul D. Hanna, Sep 03 2017

T(n,k) = Sum_{j=k..floor(n/2)}(-1)^(k+j)*C(j,k)*C(n-j,j)*Bell(n-2j).

G.f.: A(x,y) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*x + (1-y)*x^2). - Paul D. Hanna, Sep 03 2017

A250104 Triangle read by rows: T(n,k) = number of partitions of n with k circular successions (n>=0, 0 <= k <= n).

Original entry on oeis.org

0, 1, 0, 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

N. J. A. Sloane, Nov 16 2014

			Triangle begins:
0
1, 0,
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
...

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

A124323 is an essentially identical triangle, differing only in row 0 and 1.

For columns see A000296, A250105 - A250107.

t[n_, k_] := Binomial[n, k]*((-1)^(n-k)+Sum[(-1)^(j-1)*BellB[n-k-j], {j, 1, n-k}]); t[0, 0]=0; t[1, 0]=1; t[1, 1]=0; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2014 *)

A324013 Number of self-complementary set partitions of {1, ..., n} with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 4, 3, 15, 16, 75, 89, 428, 571, 2781, 4060, 20093, 31697, 159340, 268791, 1372163, 2455804, 12725447, 24012697, 126238060, 249880687, 1332071241, 2754348360, 14881206473, 32029000641, 175297058228, 391548016475, 2169832010759

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}.

			The  a(3) = 1 through a(6) = 15 self-complementary set partitions with no singletons:
  {{123}}  {{1234}}    {{12345}}    {{123456}}
           {{12}{34}}  {{135}{24}}  {{123}{456}}
           {{13}{24}}  {{15}{234}}  {{124}{356}}
           {{14}{23}}               {{1256}{34}}
                                    {{1346}{25}}
                                    {{135}{246}}
                                    {{145}{236}}
                                    {{16}{2345}}
                                    {{12}{34}{56}}
                                    {{13}{25}{46}}
                                    {{14}{25}{36}}
                                    {{15}{26}{34}}
                                    {{16}{23}{45}}
                                    {{16}{24}{35}}
                                    {{16}{25}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..500
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000296, A080107 (self-complementary), A086365, A124323, A324012 (self-conjugate).

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Count[#,{_}]==0]&]//Length,{n,0,10}]

seq(n)={my(x=x+O(x*x^(n\2)), p=exp((exp(2*x)-3)/2-x+exp(x)), q=(exp(x)-1)*p); vector(n+1, n, my(c=(n-1)\2); c!*polcoef(if(n%2, p, q), c))} \\ Andrew Howroyd, Feb 16 2022

From Andrew Howroyd, Feb 16 2022: (Start)
a(2*n) = A086365(n-1) for n > 0.
a(2*n) = n!*[x^n] exp((exp(2*x) - 3)/2 - x + exp(x));
a(2*n+1) = n!*[x^n] (exp(x) - 1)*exp((exp(2*x) - 3)/2 - x + exp(x)).
(End)

Terms a(13) and beyond from Andrew Howroyd, Feb 16 2022

A184177 Number of set partitions of {1,2,...,n} having no blocks of the form {i,i+1,i+2}.

Original entry on oeis.org

1, 1, 2, 4, 13, 46, 184, 805, 3840, 19775, 109180, 642382, 4007712, 26399764, 182939900, 1329327991, 10100670183, 80053631844, 660328296777, 5657449573120, 50255253534312, 462096537041953, 4391648990609487, 43079839534282880

Offset: 0

Emeric Deutsch, Feb 09 2011

nonn

a(n) = A184176(n,0).

			a(3) = 4 because we have 1-2-3, 1-23, 12-3, and 13-2.
a(4) = 13 because among the 15 (= Bell(4)) partitions of {1,2,3,4} only 123-4 and 1-234 have adjacent blocks of size 3.

Cf. A184176, A124323, A000296, A184174, A184175.

with(combinat): seq(add((-1)^j*binomial(n-2*j, j)*bell(n-3*j), j = 0 .. floor((1/3)*n)), n = 0 .. 23);

a(n) = Sum_{j=0..floor(n/3)} (-1)^j * binomial(n-2j,j) * Bell(n-3j,j).

A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 26, 84, 950, 6000, 62522, 556116, 6259598, 69319848, 874356338, 11384093196, 161462123894, 2397736692144, 37994808171962, 631767062124564, 11088109048500158, 203828700127054008, 3928762035148317314, 79079452776283889820, 1661265965479375937030, 36332908076071038467520, 826376466514358722894154

Offset: 0

Gus Wiseman, Feb 14 2019

nonn

			The a(4) = 2 ordered set partitions are: {{1,3},{2,4}}, {{2,4},{1,3}}.

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A000670, A001610, A032032 (adjacencies allowed), A052841 (singletons allowed), A124323, A169985, A306417, A324011 (orderless case), A324012, A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[stn]!,{stn,Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]}],{n,0,10}]

a(12)-a(26) from Alois P. Heinz, Feb 14 2019

A159830 Exponential Riordan array [exp(exp(x)-1-2x),x].

Original entry on oeis.org

1, -1, 1, 2, -2, 1, -3, 6, -3, 1, 7, -12, 12, -4, 1, -10, 35, -30, 20, -5, 1, 31, -60, 105, -60, 30, -6, 1, -21, 217, -210, 245, -105, 42, -7, 1, 204, -168, 868, -560, 490, -168, 56, -8, 1, 307, 1836, -756, 2604, -1260, 882, -252, 72, -9, 1, 2811, 3070, 9180, -2520, 6510, -2520, 1470, -360, 90, -10, 1

Offset: 0

Paul Barry, Apr 23 2009

First column is A126617. Row sums are A000296. A007318*A159830 is A124323.
The inverse is [exp(-exp(x)+1+2x),x] which has production matrix given by
1, 1,
-1, 1, 1,
-1, -2, 1, 1,
-1, -3, -3, 1, 1,
-1, -4, -6, -4, 1, 1 ...

			Triangle begins
1,
-1, 1,
2, -2, 1,
-3, 6, -3, 1,
7, -12, 12, -4, 1,
-10, 35, -30, 20, -5, 1,
31, -60, 105, -60, 30, -6, 1,
-21, 217, -210, 245, -105, 42, -7, 1,
204, -168, 868, -560, 490, -168, 56, -8, 1
Production array is
-1, 1,
1, -1, 1,
1, 2, -1, 1,
1, 3, 3, -1, 1,
1, 4, 6, 4, -1, 1,
1, 5, 10, 10, 5, -1, 1,
1, 6, 15, 20, 15, 6, -1, 1,
1, 7, 21, 35, 35, 21, 7, -1, 1,
1, 8, 28, 56, 70, 56, 28, 8, -1, 1

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[Exp[#] - 1 - 2 #]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

G.f.: 1/(1-xy+x-x^2/(1-xy-2x^2/(1-xy-x-3x^2/(1-xy-2x-4x^2/(1-... (continued fraction).

cp's OEIS Frontend

A124324 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)).

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A324011 Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A184174 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks of size 2, i.e., blocks of the form (i,i+1) (0 <= k <= floor(n/2)).

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A250104 Triangle read by rows: T(n,k) = number of partitions of n with k circular successions (n>=0, 0 <= k <= n).

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A324013 Number of self-complementary set partitions of {1, ..., n} with no singletons.

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A184177 Number of set partitions of {1,2,...,n} having no blocks of the form {i,i+1,i+2}.

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