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Number of permutations of length n with exactly 1 rising or falling succession.
11

%I #27 Sep 29 2015 15:29:59

%S 0,0,2,4,10,40,230,1580,12434,110320,1090270,11876980,141373610,

%T 1825321016,25405388150,379158271420,6039817462210,102278890975360,

%U 1834691141852174,34752142215026180,693126840194499290,14519428780464454600,318705819455462421670

%N Number of permutations of length n with exactly 1 rising or falling succession.

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

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

%H Alois P. Heinz, <a href="/A086852/b086852.txt">Table of n, a(n) for n = 0..200</a>

%H Sergey Kitaev, Jeffrey Remmel, <a href="http://arxiv.org/abs/1304.4286">(a,b)-rectangle patterns in permutations and words</a>, arXiv:1304.4286 [math.CO], 2013.

%H J. Riordan, <a href="http://projecteuclid.org/euclid.aoms/1177700181">A recurrence for permutations without rising or falling successions</a>, Ann. Math. Statist. 36 (1965), 708-710.

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

%F (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

%F a(n) ~ 2*sqrt(2*Pi)*n!/exp(2) = 0.678470495... * n!. - _Vaclav Kotesovec_, Aug 10 2013

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

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

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

%p end:

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

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Dec 21 2012

%t 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-François Alcover_, Mar 11 2014, after _Alois P. Heinz_ *)

%Y Cf. A002464, A086853, A086854, A000349, A001267.

%Y Twice A000130. A diagonal of A001100.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Aug 19 2003