A162972 -id:A162972

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A162971

Triangle read by rows: T(n,k) is number of non-derangement permutations of {1,2,...,n} having k cycles (1 <= k <= n).

+10
2

1, 0, 1, 0, 3, 1, 0, 8, 6, 1, 0, 30, 35, 10, 1, 0, 144, 210, 85, 15, 1, 0, 840, 1414, 735, 175, 21, 1, 0, 5760, 10752, 6664, 1960, 322, 28, 1, 0, 45360, 91692, 64764, 22449, 4536, 546, 36, 1, 0, 403200, 869040, 679580, 268380, 63273, 9450, 870, 45, 1, 0, 3991680, 9074736, 7704180, 3382280, 902055, 157773, 18150, 1320, 55, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Sum of entries in row n = A002467(n) (the number of non-derangement permutations of {1,2,...,n}).

T(n,2) = n*(n-2)! = A001048(n-1) for n>=3.

Sum_{k=1..n} k*T(n,k) = A162972(n).

LINKS

Alois P. Heinz, Rows n = 1..150, flattened

FORMULA

E.g.f.: G(t,z) = (1-exp(-tz))/(1-z)^t.

EXAMPLE

T(4,2) = 8 because we have (1)(234), (1)(243), (134)(2), (143)(2), (124)(3), (142)(3), (123)(4), and (132)(4).

Triangle starts:

0, 1;

0, 3, 1;

0, 8, 6, 1;

0, 30, 35, 10, 1;

0, 144, 210, 85, 15, 1;

...

MAPLE

G := (1-exp(-t*z))/(1-z)^t: Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n, t) option remember; `if`(n=0, t, add(expand((j-1)!*

b(n-j, `if`(j=1, 1, t))*x)*binomial(n-1, j-1), j=1..n))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):

seq(T(n), n=1..12); # Alois P. Heinz, Aug 15 2023

MATHEMATICA

b[n_, t_] := b[n, t] = If[n == 0, t, Sum[Expand[(j - 1)!*b[n - j, If[j == 1, 1, t]]*x]*Binomial[n - 1, j - 1], {j, 1, n}]];

T[n_] := CoefficientList[b[n, 0]/x, x];

Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 04 2024, after Alois P. Heinz *)

CROSSREFS

Cf. A001048, A002467, A162972.

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jul 22 2009

STATUS

approved

A162973

Number of cycles in all derangement permutations of {1,2,...,n}.

+10
2

0, 1, 2, 12, 64, 425, 3198, 27216, 258144, 2701737, 30933770, 384675148, 5163521856, 74417353985, 1146203362822, 18790377267840, 326682354342336, 6003886529652657, 116305541572943826, 2368629865508978284

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..449

John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.

FORMULA

a(n) = Sum_{k>=1} k*A008306(n,k).

E.g.f.: exp(-z)*(z+log(1-z))/(z-1).

a(n) ~ n! * (log(n) + gamma - 1)/exp(1), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 25 2013

a(n) = A000254(n) - A162972(n). - Anton Zakharov, Oct 18 2016

D-finite with recurrence a(n) +2*(-n+2)*a(n-1) +(n-2)*(n-6)*a(n-2) +(3*n-8)*(n-3)*a(n-3) +3*(n-3)^2*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

EXAMPLE

a(4)=12 because in the derangements of {1,2,3,4}, namely (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), and (1432), we have a total of 2+2+2+1+1+1+1+1+1=12 cycles.

MAPLE

G := exp(-z)*(z+ln(1-z))/(z-1): Gser := series(G, z = 0, 25): seq(factorial(n)*coeff(Gser, z, n), n = 1 .. 22);

MATHEMATICA

With[{nn=20}, Rest[CoefficientList[Series[Exp[-x] (x+Log[1-x])/(x-1), {x, 0, nn}], x] Range[0, nn]!]] (* Harvey P. Dale, Jul 25 2013 *)

PROG

(PARI) x='x+O('x^30); concat([0], Vec(serlaplace(exp(-x)*(x+log(1-x))/(x -1)))) \\ G. C. Greubel, Sep 01 2018