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A059438 Triangle T(n,k) (1 <= k <= n) read by rows: T(n,k) = number of permutations of [1..n] with k components. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

REFERENCES

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

LINKS

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

A. Di Crescenzo, B. Martinucci, A. 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.

P. Hegarty and A. 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

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, 1;

   3, 2, 1;

  13, 7, 3, 1;

  ...

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 *)

CROSSREFS

Diagonals give A003319, A059439, A059440, A055998.

Cf. A000007 A085771 A084938.

Sequence in context: A246381 A048647 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

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Last modified November 16 00:11 EST 2018. Contains 317252 sequences. (Running on oeis4.)