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A179455 Triangle read by rows: number of permutation trees of power n and height <= k + 1. 6
1, 1, 1, 2, 1, 5, 6, 1, 15, 23, 24, 1, 52, 106, 119, 120, 1, 203, 568, 700, 719, 720, 1, 877, 3459, 4748, 5013, 5039, 5040, 1, 4140, 23544, 36403, 39812, 40285, 40319, 40320, 1, 21147, 176850, 310851, 354391, 362057, 362836, 362879, 362880 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Partial row sums of A179454. Special cases: A179455(n,1) = BellNumber(n) = A000110(n) for n > 1; A179455(n,n-1) = n! for n > 1 and A179455(n,n-2) = A033312(n) for n > 1. Column 3 is A187761(n) for n >= 3.

See the interpretation of Joerg Arndt in A187761: Maps such that f^[k](x) = f^[k-1](x) correspond to column k of A179455 (for n >= k). - Peter Luschny, Jan 08 2013

LINKS

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

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, 2;

0, 1, 5, 6;

0, 1, 15, 23, 24;

0, 1, 52, 106, 119, 120;

0, 1, 203, 568, 700, 719, 720;

0, 1, 877, 3459, 4748, 5013, 5039, 5040;

0, 1, 4140, 23544, 36403, 39812, 40285, 40319, 40320;

0, 1, 21147, 176850, 310851, 354391, 362057, 362836, 362879, 362880;

PROG

(Sage)

# Generating algorithm from Joerg Arndt.

def A179455row(n):

    def generate(n, k):

        if n == 0 or k == 0: return 0

        for j in range(n-1, 0, -1):

            f = a[j] + 1

            while f <= j:

                a[j] = f1 = fl = f

                for i in range(k):

                    fl = f1

                    f1 = a[fl]

                if f1 == fl: return j

                f += 1

            a[j] = 0

        return 0

    count = [1 for j in range(n)] if n > 0 else [1]

    for k in range(n):

        a = [0 for j in range(n)]

        while generate(n, k) != 0:

            count[k] += 1

    return count

for n in range(9): A179455row(n) # Peter Luschny, Jan 08 2013

(Sage) # Alternatively, based on the function bell_transform defined in A264428:

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

def A179455_matrix(dim):

    b = [1]+[0]*(dim-1); L = [b]

    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][n] if k<=n else 0)

print A179455_matrix(10) # Peter Luschny, Dec 06 2015

CROSSREFS

Cf. A000110, A179454, A179456, A187761, A264428.

Sequence in context: A095801 A128567 A217204 * A039810 A124575 A178121

Adjacent sequences:  A179452 A179453 A179454 * A179456 A179457 A179458

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.