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

Author

Emeric Deutsch, Oct 28 2006

Keywords

• nonn
• tabf

Comments

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

Examples

			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;
  ...
		

Links

• 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.

Crossrefs

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

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

Programs

Maple

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

Mathematica

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 *)

Formula

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