OFFSET
0,6
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
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.
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
EXAMPLE
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;
...
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 28 2006
STATUS
approved