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A237996 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.

%I A237996 #31 May 16 2025 16:38:35
%S A237996 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 A237996 0,1624,0,175,0,1,0,0,13068,0,6769,0,322,0,1,0,40320,0,118124,0,22449,
%U A237996 0,546,0,1,0,0,1026576,0,723680,0,63273,0,870,0,1
%N A237996 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 A237996 Row sums = A001710(n).
%C A237996 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 A237996 J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 87, problem # 20.
%H A237996 Alois P. Heinz, <a href="/A237996/b237996.txt">Rows n = 0..140, flattened</a>
%H A237996 Wikipedia, <a href="https://en.wikipedia.org/wiki/Parity_of_a_permutation">Parity of a permutaion</a>
%F A237996 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 A237996 From _Peter Bala_, Jun 25 2024: (Start)
%F A237996 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 A237996 n-th row polynomial R(n, x) = (1/2)*( x*(x + 1)*...*(x + n + 1) + x*(x - 1)*...*(x - n - 1) ).
%F A237996 For n >= 1, the zeros of R(n, x) are purely imaginary. (End)
%e A237996 Triangle begins:
%e A237996   1;
%e A237996   0,     1;
%e A237996   0,     0,       1;
%e A237996   0,     2,       0,      1;
%e A237996   0,     0,      11,      0,      1;
%e A237996   0,    24,       0,     35,      0,     1;
%e A237996   0,     0,     274,      0,     85,     0,     1;
%e A237996   0,   720,       0,   1624,      0,   175,     0,   1;
%e A237996   0,     0,   13068,      0,   6769,     0,   322,   0,   1;
%e A237996   0, 40320,       0, 118124,      0, 22449,     0, 546,   0, 1;
%e A237996   0,     0, 1026576,      0, 723680,     0, 63273,   0, 870, 0, 1;
%e A237996   ...
%p A237996 with(combinat):
%p A237996 b:= proc(n, i, t) option remember; expand(`if`(n=0, t, `if`(i<1,
%p A237996       0, add(x^j*multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j,
%p A237996       i-1, irem(t+`if`(irem(i,2)=0, j, 0), 2)), j=0..n/i))))
%p A237996     end:
%p A237996 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 1)):
%p A237996 seq(T(n), n=0..12);  # _Alois P. Heinz_, Mar 09 2015
%p A237996 # Alternative:
%p A237996 A132393 := (n, k) -> abs(Stirling1(n, k)):
%p A237996 T := (n, k) -> ifelse((n::even and k::even) or (n::odd and k::odd), A132393(n, k),
%p A237996 0): seq(seq(T(n, k), k = 0..n), n = 0..9);  # _Peter Luschny_, Jun 26 2024
%t A237996 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 A237996 Cf. A008275, A130534, A132393, A164652.
%K A237996 nonn,tabl
%O A237996 0,8
%A A237996 _Geoffrey Critzer_, Feb 16 2014