

A157400


A partition product with biggestpart statistic of Stirling_1 type (with parameter k = 2) as well as of Stirling_2 type (with parameter k = 2), (triangle read by rows).


25



1, 1, 2, 1, 6, 6, 1, 24, 24, 24, 1, 80, 180, 120, 120, 1, 330, 1200, 1080, 720, 720, 1, 1302, 7770, 10920, 7560, 5040, 5040, 1, 5936, 57456, 102480, 87360, 60480, 40320, 40320, 1, 26784, 438984, 970704, 1103760, 786240, 544320, 362880, 362880
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OFFSET

1,3


COMMENTS

Partition product of Product_{j=0..n1} ((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..n2} (kn+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.
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



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..n1} (j1)
OR f_n = Product_{j=0..n2} (jn) since both have the same absolute value n!.
E.g.f. of column k: exp((x^(k+1)x)/(x1))exp((x^kx)/(x1)).  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)/(x1))exp((x^kx)/(x1)):
T:= (n, k)> n!*coeff(series(egf(k), x, n+1), x, n):


MATHEMATICA

egf[k_] := Exp[(x^(k+1)x)/(x1)]  Exp[(x^kx)/(x1)]; T[n_, k_] := n! * SeriesCoefficient[egf[k], {x, 0, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* JeanFrançois Alcover, Oct 11 2015, after Alois P. Heinz *)


CROSSREFS

Cf. A157396, A157397, A157398, A157399, A080510, A157401, A157402, A157403, A157404, A157405, A157386, A157385, A157384, A157383, A126074, A157391, A157392, A157393, A157394, A157395.


KEYWORD



AUTHOR



STATUS

approved



