OFFSET
1,5
COMMENTS
Sum of the n-th row is the number of all permutations of n elements: Sum_{k=1..n, T(n,k)} = n! = A000142(n) We can extend T(n,k)=0, if k<=0 or k>n.
From Peter Luschny, Mar 07 2009: (Start)
Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A102189.
Same partition product with length statistic is A008275.
Row sum is A000142. (End)
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.
IBM Research, Ponder This, December 2006.
Peter Luschny, Counting with Partitions. [From Peter Luschny, Mar 07 2009]
Peter Luschny, Generalized Stirling_1 Triangles. [From Peter Luschny, Mar 07 2009]
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.
FORMULA
T(n,1) = 1 T(n,2) = n! * Sum_{k=1..[n/2], (1/(k! * (2!)^k * (n-2k)!)} T(n,k) = n!/k * (1-1/(n-k)-...-1/(k+1)-1/2k), if n/3 < k <= n/2 T(n,k) = n!/k, if n/2 < k <= n T(n,n) = (n-1)! = A000142(n-1)
E.g.f. for k-th column: exp(-x^k*LerchPhi(x,1,k))*(exp(x^k/k)-1)/(1-x). - Vladeta Jovovic, Mar 03 2007
From Peter Luschny, Mar 07 2009: (Start)
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-2}(j-n+1). (End)
Sum_{k=1..n} k * T(n,k) = A028418(n). - Alois P. Heinz, May 17 2016
EXAMPLE
Triangle T(n,k) begins:
1;
1, 1;
1, 3, 2;
1, 9, 8, 6;
1, 25, 40, 30, 24;
1, 75, 200, 180, 144, 120;
1, 231, 980, 1260, 1008, 840, 720;
1, 763, 5152, 8820, 8064, 6720, 5760, 5040;
...
MAPLE
A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
add(mul(n-i, i=1..j-1)*A(n-j, k), j=1..k)))
end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Feb 11 2013
MATHEMATICA
Table[CoefficientList[ Series[(Exp[x^m/m] - 1) Exp[Sum[x^k/k, {k, 1, m - 1}]], {x, 0, 8}], x]*Table[n!, {n, 0, 8}], {m, 1, 8}] // Transpose // Grid [From Geoffrey Critzer, May 23 2009]
PROG
(Sage)
def A126074(n, k):
f = factorial(n)
P = Partitions(n, max_part=k, inner=[k])
return sum(f // p.aut() for p in P)
for n in (1..9): print([A126074(n, k) for k in (1..n)]) # Peter Luschny, Apr 17 2016
CROSSREFS
KEYWORD
AUTHOR
Dan Dima, Mar 01 2007
STATUS
approved