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A086852 Number of permutations of length n with exactly 1 rising or falling succession. 10
0, 0, 2, 4, 10, 40, 230, 1580, 12434, 110320, 1090270, 11876980, 141373610, 1825321016, 25405388150, 379158271420, 6039817462210, 102278890975360, 1834691141852174, 34752142215026180, 693126840194499290, 14519428780464454600, 318705819455462421670 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Permutations of 12...n such that exactly one of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

S. Kitaev, J. Remmel, (a,b)-rectangle patterns in permutations and words

J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

FORMULA

Coefficient of t^1 in S[n](t) defined in A002464.

(3-n)*a(n) +(n+1)*(n-3)*a(n-1) -(n^2-4*n+5)*a(n-2) -(n-1)*(n-5)*a(n-3) +(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 06 2013

a(n) ~ 2*sqrt(2*Pi)*n!/exp(2) = 0.678470495... * n!. - Vaclav Kotesovec, Aug 10 2013

MAPLE

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]

       [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)

       -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))

    end:

a:= n-> coeff(S(n), t, 1):

seq (a(n), n=0..30);  # Alois P. Heinz, Dec 21 2012

MATHEMATICA

S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(n+1-t)*S[n-1]-(1-t)*(n-2+3*t)*S[n-2]-(1-t)^2*(n-5+t)*S[n-3]+(1-t)^3*(n-3)*S[n-4]]]; a[n_] := Coefficient[S[n], t, 1]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Mar 11 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A002464, A086853, A086854, A000349, A001267. Twice A000130. A diagonal of A001100.

Sequence in context: A108801 A193675 A111022 * A084737 A153757 A159860

Adjacent sequences:  A086849 A086850 A086851 * A086853 A086854 A086855

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 19 2003

STATUS

approved

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Last modified November 20 15:32 EST 2014. Contains 249751 sequences.