login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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

%I #25 Oct 07 2024 01:25:27

%S 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,

%T 175,21,1,0,5760,10752,6664,1960,322,28,1,0,45360,91692,64764,22449,

%U 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

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

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

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

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

%H Alois P. Heinz, <a href="/A162971/b162971.txt">Rows n = 1..150, flattened</a>

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

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

%e Triangle starts:

%e 1;

%e 0, 1;

%e 0, 3, 1;

%e 0, 8, 6, 1;

%e 0, 30, 35, 10, 1;

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

%e ...

%p 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

%p # second Maple program:

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

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

%p end:

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

%p seq(T(n), n=1..12); # _Alois P. Heinz_, Aug 15 2023

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

%t Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Apr 04 2024, after _Alois P. Heinz_ *)

%Y Cf. A001048, A002467, A162972.

%K nonn,tabl

%O 1,5

%A _Emeric Deutsch_, Jul 22 2009