OFFSET
1,3
COMMENTS
Partition product of Product_{j=0..n-1} ((k+1)*j - 1) and n! at k = -2, summed over parts with equal biggest part (Stirling_2 type) as well as partition product of Product_{j=0..n-2} (k-n+j+2) and n! at k = -2 (Stirling_1 type).
It shares this property with the signless Lah numbers.
Underlying partition triangle is A130561.
Same partition product with length statistic is A105278.
Row sum is A000262.
T(n,k) is the number of nilpotent elements in the symmetric inverse semigroup (partial bijections) on [n] having index k. Equivalently, T(n,k) is the number of directed acyclic graphs on n labeled nodes with every node having indegree and outdegree at most one and the longest path containing exactly k nodes. - Geoffrey Critzer, Nov 21 2021
LINKS
Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.
Peter Luschny, Generalized Stirling_2 Triangles.
FORMULA
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n.
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that
1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-j-1)
OR f_n = Product_{j=0..n-2} (j-n) since both have the same absolute value n!.
E.g.f. of column k: exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)). - Alois P. Heinz, Oct 10 2015
EXAMPLE
Triangle starts:
1;
1, 2;
1, 6, 6;
1, 24, 24, 24;
1, 80, 180, 120, 120;
1, 330, 1200, 1080, 720, 720;
...
MAPLE
egf:= k-> exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)):
T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Oct 10 2015
MATHEMATICA
egf[k_] := Exp[(x^(k+1)-x)/(x-1)] - Exp[(x^k-x)/(x-1)]; T[n_, k_] := n! * SeriesCoefficient[egf[k], {x, 0, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Oct 11 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Mar 09 2009, Mar 14 2009
STATUS
approved