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
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