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

0, 0, 2, 4, 10, 40, 230, 1580, 12434, 110320, 1090270, 11876980, 141373610, 1825321016, 25405388150, 379158271420, 6039817462210, 102278890975360, 1834691141852174, 34752142215026180, 693126840194499290, 14519428780464454600, 318705819455462421670

Offset: 0

Author

N. J. A. Sloane, Aug 19 2003

Keywords

• nonn
• easy

Comments

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

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

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
• Sergey Kitaev, Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
• J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.
• D. P. Robbins, The probability that neighbors remain neighbors after random rearrangements, Amer. Math. Monthly 87 (1980), 122-124.

Crossrefs

Cf. A002464, A086853, A086854, A000349, A001267, A383857.

Twice A000130. A diagonal of A001100.

Cf. A104698, A384494.

Programs

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-François Alcover, Mar 11 2014, after Alois P. Heinz *)

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

From Wolfdieter Lang, May 31 2025: (Start)

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).

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)