A292897 -id:A292897 - cp's OEIS frontend

A259834 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals n-1.

0, 0, 1, 2, 5, 20, 97, 574, 3973, 31520, 281825, 2803418, 30704101, 367114252, 4757800705, 66432995030, 994204132517, 15875195019224, 269397248811073, 4841453414347570, 91856764780324165, 1834779993945449348, 38485629141294791201, 845788826477292504302

Offset: 0

Alois P. Heinz, Jul 06 2015

a(n) counts permutations p of [n] such that p(i) <> i and (p(1) = n or p(n) = 1).

			a(2) = 1: 21.
a(3) = 2: 231, 312.
a(4) = 5: 2341, 3421, 4123, 4312, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..450

A diagonal of A259784.

Cf. A000142, A000166, A000255, A292897, A292898, A306455.

a:= proc(n) option remember; `if`(n<3, [0, 0, 1][n+1],
     ((2*n^2-11*n+13)*a(n-1) +(2*n-5)*(n-3)*a(n-2))/(2*n-7))
    end:
seq(a(n), n=0..30);

Join[{0, 0}, Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == (n - m)*y[m] + (n - m) y[m - 1], y[0] == 0, y[1] == 1, y[2] == 1}]][n], {n, 2, 30}]] (* Benedict W. J. Irwin, Nov 03 2016 *)
Table[If[n<2, 0, Subfactorial[n-2]+2*Subfactorial[n-1]], {n,0,23}] (* Peter Luschny, Oct 04 2017 *)

def A259834_list(len):
    L, u, x, y = [0], 1, 0, 0
    for n in range(len):
        y, x, u = x, x*n + u, -u
        L.append(y + 2*x)
    L[1] = 0
    return L
print(A259834_list(23)) # Peter Luschny, Oct 04 2017

a(n) = ((2*n^2-11*n+13)*a(n-1) + (2*n-5)*(n-3)*a(n-2))/(2*n-7) for n > 2.
a(n) = (n-2)! * [x^(n-2)] exp(-x)*(x+1)/(x-1)^2 for n > 1.
a(n) ~ 2 * (n-1)! / exp(1). - Vaclav Kotesovec, Jul 07 2015
a(n) = y(n,n), n > 1, where y(m+1,n) = (n-m)*y(m,n) + (n-m)*y(m-1,n), with y(0,n)=0, y(1,n)=y(2,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
From Peter Luschny, Oct 05 2017: (Start)
a(n) = (Gamma(n-1, -1) + 2*Gamma(n, -1)) / e for n >= 2.
a(n) = A000166(n-2) + 2*A000166(n-1) for n >= 2.
a(n) = (2*n-1)*A000166(n-2) - 2*(-1)^n for n >= 2.
a(n) = A000255(n-2) + A000166(n-1) for n >= 2.
a(n+2) = (-1)^n*(-hypergeom([1,1-n], [], 1) + hypergeom([2,2-n], [], 1)) = A292898(2, n) for n >= 0.
a(n) ~ 2*sqrt(2*Pi)*exp(-n-1)*n^(n-1/2). (End)
a(n+2) = A306455(n) + n! for n >= 0. - Alois P. Heinz, Feb 16 2019

A292898 Array read by ascending antidiagonals, A(m, n) = Sum_{k=1..m}(-1)^(k-n-m)* hypergeom([k, k-n-m], [], 1) for m>=1 and n>=0.

Original entry on oeis.org

1, 1, 0, 3, 2, 1, 8, 7, 5, 2, 31, 30, 27, 20, 9, 147, 146, 142, 129, 97, 44, 853, 852, 847, 826, 755, 574, 265, 5824, 5823, 5817, 5786, 5652, 5187, 3973, 1854, 45741, 45740, 45733, 45690, 45463, 44462, 40923, 31520, 14833

Offset: 0

Peter Luschny, Oct 05 2017

			Array starts:
[m\n]   0       1      2       3        4         5          6
-------------------------------------------------------------------
[1]     1,      0,     1,      2,       9,       44,       265, ...  [A000166]
[2]     1,      2,     5,     20,      97,      574,      3973, ...  [A259834(n+2)]
[3]     3,      7,    27,    129,     755,     5187,     40923, ...  [A292897]
[4]     8,     30,   142,    826,    5652,    44462,    394970, ...
[5]    31,    146,   847,   5786,   45463,   403514,   3990679, ...
[6]   147,    852,  5817,  45690,  405423,  4008768,  43692933, ...
[7]   853,   5823, 45733, 405779, 4012101, 43727687, 520723477, ...
  A003470,A193464,A293295.
Displayed as a triangle:
[1]     1;
[2]     1,      0;
[3]     3,      2,     1;
[4]     8,      7,     5,    2;
[5]    31,     30,    27,   20,    9;
[6]   147,    146,   142,  129,   97,   44;
[7]   853,    852,   847,  826,  755,  574,  265;
[8]  5824,   5823,  5817, 5786, 5652, 5187, 3973, 1854;
  A003470,A193464,A293295.
This triangle has row sums A193463.

Cf. A000166, A003470, A193463, A193464, A259834, A292897, A293295.

A := (m, n) -> add((-1)^(k-n-m)*hypergeom([k, k-n-m], [], 1), k=1..m):
seq(lprint(seq(simplify(A(m, n)), n=0..6)), m=1..7);

A[m_, n_] :=  Sum[(-1)^(k-n-m) HypergeometricPFQ[{k, k-n-m},{}, 1], {k, 1, m} ];
Table[Table[A[m, n], {n,0,6}], {m,1,7}]

cp's OEIS Frontend

A259834 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula