login
Triangular array read by rows. T(n,k) is the number of even permutations of {1,2,...,n} that have exactly k cycles, n >= 0, 0 <= k <= n.
1

%I #25 Jun 26 2024 14:29:39

%S 1,0,1,0,0,1,0,2,0,1,0,0,11,0,1,0,24,0,35,0,1,0,0,274,0,85,0,1,0,720,

%T 0,1624,0,175,0,1,0,0,13068,0,6769,0,322,0,1,0,40320,0,118124,0,22449,

%U 0,546,0,1,0,0,1026576,0,723680,0,63273,0,870,0,1

%N Triangular array read by rows. T(n,k) is the number of even permutations of {1,2,...,n} that have exactly k cycles, n >= 0, 0 <= k <= n.

%C Row sums = A001710(n).

%C If a permutation of {1, 2, ..., n} is written as a product of m disjoint cycles (where the fixed points of the permutation are viewed as 1-cycles) then the parity of the permutation is (-1)^(n-m). It is an even permutation if the number of cycles of even length is even (possibly zero), and it is an odd permutation if the number of cycles of even length is odd. - _Peter Bala_, Jun 25 2024

%D J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 87, problem # 20.

%H Alois P. Heinz, <a href="/A237996/b237996.txt">Rows n = 0..140, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Parity_of_a_permutation">Parity of a permutaion</a>

%F E.g.f.: exp(y*A(x))*cosh(y*B(x)) where A(x)= log((1 + x)/(1 - x))^(1/2) and B(x)=log(1/(1-x^2)^(1/2).

%F From _Peter Bala_, Jun 25 2024: (Start)

%F If n and k are both even or both odd, then T(n, k) is equal to the Stirling cycle number |s(n, k)| = A132393(n, k), and 0 otherwise.

%F n-th row polynomial R(n, x) = (1/2)*( x*(x + 1)*...*(x + n + 1) + x*(x - 1)*...*(x - n - 1) ).

%F For n >= 1, the zeros of R(n, x) are purely imaginary. (End)

%e 1,

%e 0, 1,

%e 0, 0, 1,

%e 0, 2, 0, 1,

%e 0, 0, 11, 0, 1,

%e 0, 24, 0, 35, 0, 1,

%e 0, 0, 274, 0, 85, 0, 1,

%e 0, 720, 0, 1624, 0, 175, 0, 1,

%e 0, 0, 13068, 0, 6769, 0, 322, 0, 1,

%e 0, 40320, 0, 118124, 0, 22449, 0, 546, 0, 1,

%e 0, 0, 1026576, 0, 723680, 0, 63273, 0, 870, 0, 1

%p with(combinat):

%p b:= proc(n, i, t) option remember; expand(`if`(n=0, t, `if`(i<1,

%p 0, add(x^j*multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j,

%p i-1, irem(t+`if`(irem(i,2)=0, j, 0), 2)), j=0..n/i))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 1)):

%p seq(T(n), n=0..12); # _Alois P. Heinz_, Mar 09 2015

%p # Alternative:

%p A132393 := (n, k) -> abs(Stirling1(n, k)):

%p T := (n, k) -> ifelse((n::even and k::even) or (n::odd and k::odd), A132393(n, k),

%p 0): seq(seq(T(n, k), k = 0..n), n = 0..9); # _Peter Luschny_, Jun 26 2024

%t nn=11;a=Log[((1+x)/(1-x))^(1/2)];b=Log[1/(1-x^2)^(1/2)];Table[Take[(Range[0,nn]!CoefficientList[Series[Exp[y a]Cosh[y b] ,{x,0,nn}],{x,y}])[[n]],n],{n,1,nn}]//Grid

%Y Cf. A008275, A130534, A132393, A164652.

%K nonn,tabl

%O 0,8

%A _Geoffrey Critzer_, Feb 16 2014