A059438 Triangle T(n,k) (1 <= k <= n) read by rows: T(n,k) is the number of permutations of [1..n] with k components.

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 32, 12, 4, 1, 461, 177, 58, 18, 5, 1, 3447, 1142, 327, 92, 25, 6, 1, 29093, 8411, 2109, 531, 135, 33, 7, 1, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1, 2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1

Offset

1,4

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).

Antonio Di Crescenzo, Barbara Martinucci, and Abdelaziz Rhandi, A linear birth-death process on a star graph and its diffusion approximation, arXiv:1405.4312 [math.PR], 2014.

FindStat - Combinatorial Statistic Finder, The decomposition number of a permutation.

Peter Hegarty and Anders Martinsson, On the existence of accessible paths in various models of fitness landscapes, arXiv:1210.4798 [math.PR], 2012-2014. - From N. J. A. Sloane, Jan 01 2013

Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.

Formula

Let f(x) = Sum_{n >= 0} n!*x^n, g(x) = 1 - 1/f(x). Then g(x) is g.f. for first diagonal A003319 and Sum_{n >= k} T(n, k)*x^n = g(x)^k.

Triangle T(n, k), n > 0 and k > 0, read by rows; given by [0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.

T(n+k, k) = Sum_{a_1 + a_2 + ... + a_k = n} A003319(a_1)*A003319(a_2)*...*A003319(a_k). T(n, k) = 0 if n < k, T(n, 1) = A003319(n) and for n >= k T(n, k)= Sum_{j>=1} T(n-j, k-1)* A003319(j). A059438 is formed from the self convolution of its first column (A003319). - Philippe Deléham, Feb 04 2004

Sum_{k>0} T(n, k) = n!; see A000142. - Philippe Deléham, Feb 05 2004

If g(x) = x + x^2 + 3*x^3 + 13*x^4 + ... is the generating function for the number of permutations with no global descents, then 1/(1-g(x)) is the generating function for n!. Setting t=1 in f(x, t) implies Sum_{k=1..n} T(n,k) = n!. Let g(x) be the o.g.f. for A003319. Then the o.g.f. for this table is given by f(x, t) = 1/(1 - t*g(x)) - 1 (i.e., the coefficient of x^n*t^k in f(x,t) is T(n,k)). - Mike Zabrocki, Jul 29 2004

Example

Triangle begins:
[1] [     1]
[2] [     1,     1]
[3] [     3,     2,     1]
[4] [    13,     7,     3,    1]
[5] [    71,    32,    12,    4,   1]
[6] [   461,   177,    58,   18,   5,   1]
[7] [  3447,  1142,   327,   92,  25,   6,  1]
[8] [ 29093,  8411,  2109,  531, 135,  33,  7, 1]
[9] [273343, 69692, 15366, 3440, 800, 188, 42, 8, 1]

Maple

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, A003319); # Peter Luschny, Oct 09 2022

Mathematica

(* p = indecomposable permutations = A003319 *) p[n_] := p[n] = n! - Sum[ k!*p[n-k], {k, 1, n-1}]; t[n_, k_] /; n < k = 0; t[n_, 1] := p[n]; t[n_, k_] /; n >= k := t[n, k] = Sum[ t[n-j, k-1]*p[j], {j, 1, n}]; Flatten[ Table[ t[n, k], {n, 1, 10}, {k, 1, n}] ] (* Jean-François Alcover, Mar 06 2012, after Philippe Deléham *)

Prog

(SageMath)
def A059438_triangle(dim) :
    R = PolynomialRing(ZZ, 'x')
    C = [R(0)] + [R(1) for i in range(dim+1)]
    A = [(i + 2) // 2 for i in range(dim+1)]
    A[0] = R.gen(); T = []
    for k in range(1, dim+1) :
        for n in range(k, 0, -1) :
            C[n] = C[n-1] + C[n+1] * A[n-1]
        T.append(list(C[1])[1::])
    return T
A059438_triangle(8) # Peter Luschny, Sep 10 2022
(SageMath) Alternatively, using the function PartTrans from A357078:
# Adds a (0, 0)-based column (1, 0, 0, ...) to the left of the triangle.
dim = 10
A = ZZ[['t']]; g = A([0]+[factorial(n) for n in range(1, 30)]).O(dim+2)
PartTrans(dim, lambda n: list(g / (1 +  g))[n]) # Peter Luschny, Sep 11 2022

Crossrefs

A version with reflected rows is A263484.

Diagonals give A003319, A059439, A059440, A055998.

Cf. A000007, A085771, A084938.

T(2n,n) gives A308650.

Sequence in context: A048647 A358310 A180190 * A156628 A104980 A316566

Adjacent sequences: A059435 A059436 A059437 * A059439 A059440 A059441

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane, Feb 01 2001

Extensions

More terms from Vladeta Jovovic, Mar 04 2001

Status

approved