A145222 -id:A145222 - cp's OEIS frontend

A145225 Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 6, 0, 6, 0, 0, 20, 30, 0, 10, 0, 0, 135, 120, 90, 0, 15, 0, 0, 924, 945, 420, 210, 0, 21, 0, 0, 7420, 7392, 3780, 1120, 420, 0, 28, 0, 0, 66744, 66780, 33264, 11340, 2520, 756, 0, 36, 0, 0

Offset: 0

Abdullahi Umar, Oct 10 2008

			Triangle starts:
   0;
   0,  0;
   1,  0, 0;
   0,  3, 0,  0;
   6,  0, 6,  0, 0;
  20, 30, 0, 10, 0, 0;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Row sums are A001710 for n > 1.
Columns k=0..2 are A000387, A145222, A145223.
Cf. A000387, A008290, A055137, A145224.

A145225 := proc(n,k)
    binomial(n,k)*A000387(n-k) ; # re-use code of A000387
end proc:
seq(seq(A145225(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 06 2023

A145225[n_, k_] := Binomial[n, k]*Binomial[n - k, 2]*Subfactorial[n - k - 2];
Table[A145225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 31 2025 *)

T(n,k) = C(n,k) * A000387(n-k).
E.g.f.: x^(k+2) * exp(-x) / (2*k!*(1-x)).
T(n,k) + A145224(n,k) = A008290(n,k). - R. J. Mathar, Jul 06 2023
T(n,k) = (A008290(n,k) - A055137(n,k)) / 2. - Julian Hatfield Iacoponi, Aug 08 2024

A145219 a(n) is the number of even permutations (of an n-set) with exactly 1 fixed point.

Original entry on oeis.org

1, 0, 0, 8, 15, 144, 910, 7440, 66717, 667520, 7342236, 88107480, 1145396395, 16035550608, 240533257770, 3848532125984, 65425046139705, 1177650830517120, 22375365779822392, 447507315596451240, 9397653627525472071, 206748379805560390160, 4755212735527888968390

Offset: 1

Abdullahi Umar, Oct 09 2008

nonn

			a(4) = 8 because there are exactly 8 even permutations (of a 4-set) having 1 fixed point, namely: (123), (132), (124), (142), (134), (143), (234), (243).

Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Cf. A003221, A145220, A145222.

x = 'x + O('x^30); Vec(serlaplace((x*(1-x^2/2) * exp(-x))/(1-x))) \\ Michel Marcus, Apr 04 2016

a(n) = n*A003221(n-1), (n > 0).
E.g.f.: (x*(1-x^2/2) * exp(-x))/(1-x).
D-finite with recurrence (-3*n+7)*a(n) +(n-2)*(3*n-10)*a(n-1) +(9*n-8)*(n-3)*a(n-2) +2*(3*n-2)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 06 2023

More terms from Alois P. Heinz, Nov 19 2013

A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

Original entry on oeis.org

0, 0, 6, 0, 90, 420, 3780, 33264, 333900, 3670920, 44054010, 572697840, 8017775766, 120266628300, 1924266063720, 32712523068960, 588825415259640, 11187682889909904, 223753657798227150, 4698826813762734240, 103374189902780197170, 2377606367763944481780

Offset: 2

Abdullahi Umar, Oct 09 2008

nonn

			a(4) = 6 because there are exactly 6 odd permutations (of a 4-set) having 2 fixed points, namely: (12), (13), (14), (23), (24), (34).

Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Cf. A000387 (odd permutations with no fixed points), A145222 (odd permutations with exactly 1 fixed point), A145220 (even permutations with exactly 2 fixed points).

egf:= x^4 * exp(-x)/(4*(1-x));
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=2..30);  # Alois P. Heinz, Feb 01 2011

A000387[n_] := Subfactorial[n-2] Binomial[n, 2];
a[n_] := (n(n-1)/2) A000387[n-2];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 30 2025 *)

x = 'x + O('x^30); Vec(serlaplace(((x^4)*exp(-x))/(4*(1-x)))) \\ Michel Marcus, Apr 04 2016

a(n) = A145225(n,2) = (n*(n-1)/2) * A000387(n-2), (n > 1).
E.g.f.: x^4*exp(-x)/(4*(1-x)).
D-finite with recurrence +(-n+6)*a(n) +(n-2)*(n-7)*a(n-1) +(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

More terms from Alois P. Heinz, Feb 01 2011

cp's OEIS Frontend

A145225 Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

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A145219 a(n) is the number of even permutations (of an n-set) with exactly 1 fixed point.

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A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

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