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A001251 Number of permutations of order n with the length of longest run equal 3.
(Formerly M2031 N0803)
12
0, 0, 2, 12, 70, 442, 3108, 24216, 208586, 1972904, 20373338, 228346522, 2763212980, 35926266244, 499676669254, 7405014187564, 116511984902094, 1940073930857802, 34087525861589564, 630296344519286304, 12235215845125112122, 248789737587365945992 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Terms for n>=13 are incorrect).

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

Max Alekseyev and 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 runs length, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From N. J. A. Sloane, Oct 23 2012

Richard Ehrenborg and JiYoon Jung, Descent pattern avoidance, arXiv preprint:1312.2027 [math.CO], 6 Dec 2013.

FORMULA

a(n) ~ c * d^n * n!, where d = 0.92403585760753647721113386869798700855648617941... is the root of the equation 8 - 2*sin(sqrt(phi)/d) * (2*sqrt(5*(phi-1)) * cosh(sqrt(phi-1)/d) + 2*sinh(sqrt(phi-1)/d)) + 2*cos(sqrt(phi)/d) * (6*cosh(sqrt(phi-1)/d) + 2*sqrt(5*phi) * sinh(sqrt(phi-1)/d)) = 0, phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 1.259371257828351725264434486385284120241474052544197367866029465830756911... - Vaclav Kotesovec, Sep 06 2014, updated Aug 18 2018

MATHEMATICA

length = 3;

g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[t<length, Sum[g[u + j - 1, o - j, t+1], {j, 1, o}], 0]];

b[u_, o_, t_] := b[u, o, t] = If[t == length, g[u, o, t], Sum[b[o + j - 1, u - j, 2], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];

a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];

Table[a[n], {n, 1, 30}] (* Jean-Fran├žois Alcover, Aug 18 2018, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A116398 A001542 A059229 * A143357 A012426 A012421

Adjacent sequences:  A001248 A001249 A001250 * A001252 A001253 A001254

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected and extended by Max Alekseyev at suggestion of Sean A. Irvine, May 04 2012

STATUS

approved

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Last modified December 16 15:11 EST 2018. Contains 318172 sequences. (Running on oeis4.)