A373339 - OEIS

%I #22 Jun 28 2024 22:51:45

%S 1,1,1,1,4,36,296,2360,19776,180544,1812352,19953792,239490560,

%T 3113487872,43589096448,653837077504,10461394714624,177843713556480,

%U 3201186851815424,60822550202187776,1216451004083601408,25545471085844758528,562000363888782868480

%N Number of permutations in symmetric group S_n with an even number of cycles of length 2 or more.

%F a(n) = n!/2 - (n-2)*2^(n-2) = A001710(n) - A036289(n-2).

%F a(n) = A000142(n) - A373340(n).

%F E.g.f.: (1/(1 - x) + exp(2*x)*(1 - x))/2. - _Stefano Spezia_, Jun 05 2024

%e a(1)=a(2)=a(3)=1 due to S_1,S_2,S_3 containing 1 permutation with an even number of non-fixed point cycles: the identity permutation, with 0 non-fixed point cycles.

%e a(4)=4 due to S_4 containing 4 permutations with an even number of non-fixed point cycles: the 3 (2,2)-cycles (12)(34),(13)(24),(14)(23); and the identity permutation (1)(2)(3)(4).

%o (PARI) a(n) = n!/2 - (n-2)*2^(n-2); \\ _Michel Marcus_, Jun 05 2024

%Y Cf. A373340 (odd case), A000142, A001710, A036289.

%Y Row sums of triangle A373417.

%K nonn

%O 0,5

%A _Julian Hatfield Iacoponi_, Jun 01 2024