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A179454 Permutation trees of power n and height k. 5
1, 1, 1, 1, 1, 4, 1, 1, 14, 8, 1, 1, 51, 54, 13, 1, 1, 202, 365, 132, 19, 1, 1, 876, 2582, 1289, 265, 26, 1, 1, 4139, 19404, 12859, 3409, 473, 34, 1, 1, 21146, 155703, 134001, 43540, 7666, 779, 43, 1, 1, 115974, 1335278, 1471353, 569275, 120200, 15456, 1209, 53, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

A permutation tree is a labeled rooted tree that has vertex set {0,1,2,..,n} and root 0 in which each child is larger than its parent and the children are in ascending order from the left to the right. The height of a permutation tree is the number of descendants of the root on the longest chain starting at the root and ending at a leaf. This defines C(n,height) for 1<=height<=n. Row sum is n!.

Setting T(n,k) = C(n,k+1) for 0<=k<n and additionally T(0,0) = 1 the T(n,k) can also be interpreted as coefficients of polynomials P_n(x) = Sum_{k=0..n-1} T(n,k) x^k. An analogous construction classifying permutation trees by width (number of leafs) gives the Eulerian numbers as defined in A008292 and the Eulerian polynomials as defined via DLMF 26.14.1. (See A123125 for the triangle with an (0,0)-based offset.)

LINKS

Alois P. Heinz, Rows n = 0..141, flattened

FindStat - Combinatorial Statistic Finder, The height index of a permutation.

Peter Luschny, Permutation Trees.

EXAMPLE

As a (0,0)-based triangle with an additional column [1,0,0,0,...] at the left hand side:

[ 1 ]

[ 0, 1 ]

[ 0, 1,    1 ]

[ 0, 1,    4,     1 ]

[ 0, 1,   14,     8,     1 ]

[ 0, 1,   51,    54,    13,    1 ]

[ 0, 1,  202,   365,   132,   19,   1 ]

[ 0, 1,  876,  2582,  1289,  265,  26,  1 ]

[ 0, 1, 4139, 19404, 12859, 3409, 473, 34, 1]

--------------------------------------------

The height statistic over permutations, n=4.

[1, 2, 3, 4] => 2; [1, 2, 4, 3] => 3; [1, 3, 2, 4] => 3; [1, 3, 4, 2] => 3;

[1, 4, 2, 3] => 3; [1, 4, 3, 2] => 4; [2, 1, 3, 4] => 2; [2, 1, 4, 3] => 3;

[2, 3, 1, 4] => 2; [2, 3, 4, 1] => 2; [2, 4, 1, 3] => 2; [2, 4, 3, 1] => 3;

[3, 1, 2, 4] => 2; [3, 1, 4, 2] => 2; [3, 2, 1, 4] => 2; [3, 2, 4, 1] => 2;

[3, 4, 1, 2] => 2; [3, 4, 2, 1] => 3; [4, 1, 2, 3] => 1; [4, 1, 3, 2] => 2;

[4, 2, 1, 3] => 2; [4, 2, 3, 1] => 2; [4, 3, 1, 2] => 2; [4, 3, 2, 1] => 3;

Gives row(4) = [0, 1, 14, 8, 1]. - Peter Luschny, Dec 09 2015

MAPLE

b:= proc(n, t, h) option remember; `if`(n=0 or h=0, 1, add(

      binomial(n-1, j-1)*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n))

    end:

T:= (n, k)-> b(n, 1, k-1)-`if`(k<2, 0, b(n, 1, k-2)):

seq(seq(T(n, k), k=min(n, 1)..n), n=0..12);  # Alois P. Heinz, Aug 24 2017

PROG

(Sage)

# The function bell_transform is defined in A264428.

# Adds the column (1, 0, 0, 0, ..) to the left hand side and starts at n=0.

def A179454_matrix(dim):

    a = [2]+[0]*(dim-1); b = [1]+[0]*(dim-1); L = [b, a]

    for k in range(dim):

        b = [sum((bell_transform(n, b))) for n in range(dim)]

        L.append(b)

    return matrix(ZZ, dim, lambda n, k: L[k+1][n]-L[k][n] if k<=n else 0)

A179454_matrix(9) # Peter Luschny, Dec 07 2015

(Sage) # Alternatively, based on FindStat statistic St000308:

def statistic_000308(pi):

    if pi == []: return 0

    h, i, branch, next = 0, len(pi), [0], pi[0]

    while true:

        while next < branch[len(branch)-1]:

            del(branch[len(branch)-1])

        current = 0

        while next > current:

            i -= 1

            branch.append(next)

            h = max(h, len(branch)-1)

            if i == 0: return h

            current, next = next, pi[i]

def A179454_row(n):

    L = [0]*(n+1)

    for p in Permutations(n):

        L[statistic_000308(p)] += 1

    return L

[A179454_row(n) for n in range(8)] # Peter Luschny, Dec 09 2015

CROSSREFS

Cf. A008292, A123125, A179455, A179456, A264428.

Row sums give A000142.

Sequence in context: A064281 A267318 A050154 * A058711 A202906 A177984

Adjacent sequences:  A179451 A179452 A179453 * A179455 A179456 A179457

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Aug 11 2010

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.