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A000456 Number of permutations of [n] in which the longest increasing run has length 5.
(Formerly M4735 N2027)
6
0, 0, 0, 0, 1, 10, 99, 1024, 11304, 133669, 1695429, 23023811, 333840443, 5153118154, 84426592621, 1463941342191, 26793750988542, 516319125748337, 10451197169218523, 221738082618710329, 4921234092461339819, 114041894068935641488 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from Max Alekseyev)

Max A. Alekseyev, On the number of permutations with bounded run lengths, arXiv preprint arXiv:1205.4581, 2012.

EXAMPLE

a(6)=10 because we have (12346)5, (12356)4, (12456)3, (13456)2, (23456)1, 6(12345), 5(12346), 4(12356), 3(12456) and 2(13456), where the parentheses surround increasing runs of length 5.

MATHEMATICA

b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]]; T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1]; a[n_] := T[n, 5]; Array[a, 25] (* Jean-Fran├žois Alcover, Feb 08 2016, after Alois P. Heinz in A008304 *)

CROSSREFS

Column 5 of A008304. Other columns: A000303, A000402, A000434, A000467.

Cf. A001250, A001251, A001252, A001253, A010026, A211318.

Sequence in context: A179557 A300000 A213454 * A138365 A190823 A187019

Adjacent sequences:  A000453 A000454 A000455 * A000457 A000458 A000459

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Better description from Emeric Deutsch, May 08 2004

Edited and extended by Max Alekseyev, May 20 2012

STATUS

approved

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Last modified March 29 03:09 EDT 2020. Contains 333104 sequences. (Running on oeis4.)