cp's OEIS Frontend

A086852 Number of permutations of length n with exactly 1 rising or falling succession.

%I A086852 #38 Jun 22 2025 00:16:40
%S A086852 0,0,2,4,10,40,230,1580,12434,110320,1090270,11876980,141373610,
%T A086852 1825321016,25405388150,379158271420,6039817462210,102278890975360,
%U A086852 1834691141852174,34752142215026180,693126840194499290,14519428780464454600,318705819455462421670
%N A086852 Number of permutations of length n with exactly 1 rising or falling succession.
%C A086852 Permutations of 12...n such that exactly one of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
%C A086852 For the number of such permutations without (n-1)n or n(n-1) see A383857(n), for n >= 1. - _Wolfdieter Lang_, May 22 2025
%D A086852 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
%H A086852 Alois P. Heinz, <a href="/A086852/b086852.txt">Table of n, a(n) for n = 0..200</a>
%H A086852 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 A086852 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.
%H A086852 D. P. Robbins, <a href="http://www.jstor.org/stable/2321990">The probability that neighbors remain neighbors after random rearrangements</a>, Amer. Math. Monthly 87 (1980), 122-124.
%F A086852 Coefficient of t^1 in S[n](t) defined in A002464.
%F A086852 (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 A086852 a(n) ~ 2*sqrt(2*Pi)*n!/exp(2) = 0.678470495... * n!. - _Vaclav Kotesovec_, Aug 10 2013
%F A086852 From _Wolfdieter Lang_, May 31 2025: (Start)
%F A086852 a(n) = Sum_{i=1..n-1} (-1)^(i-1)*i*(n-i)!*Sum_{j=1..i} 2^j*binomial(i-1, j-1)*binomial(n-i, j), for n >= 0. See the D. P. Robbins link, p. 123, eq. (7), A(n, 1).
%F A086852 a(n+2) = Sum_{k=0..n} 2*R(n, k)*B(n, k), with B(n, k) = A384494(n, k) = (-1)^k*(k+1)*(n+1-k)!, and R(n, k) = A104698(n, k), for n >= 0. This equals (2*MR*MB^t)_{n,n}, with the (infinite) square matrices with vanishing upper diagonals corresponding to R and B, and t indicates transposition. (End)
%p A086852 S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
%p A086852        [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
%p A086852        -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
%p A086852     end:
%p A086852 a:= n-> coeff(S(n), t, 1):
%p A086852 seq(a(n), n=0..30);  # _Alois P. Heinz_, Dec 21 2012
%t A086852 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 A086852 Cf. A002464, A086853, A086854, A000349, A001267, A383857.
%Y A086852 Twice A000130. A diagonal of A001100.
%Y A086852 Cf. A104698, A384494.
%K A086852 nonn,easy
%O A086852 0,3
%A A086852 _N. J. A. Sloane_, Aug 19 2003