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A374419 Triangle read by rows: T(n,k) = number of permutations in symmetric group S_n with an even number of non-fixed point cycles, without k<=n particular fixed points.

%I A374419 #23 Jul 31 2024 11:55:15
%S A374419 1,1,0,1,0,0,1,0,0,0,4,3,3,3,3,36,32,29,26,23,20,296,260,228,199,173,
%T A374419 150,130,2360,2064,1804,1576,1377,1204,1054,924,19776,17416,15352,
%U A374419 13548,11972,10595,9391,8337,7413,180544,160768,143352,128000,114452,102480,91885,82494,74157,66744
%N A374419 Triangle read by rows: T(n,k) = number of permutations in symmetric group S_n with an even number of non-fixed point cycles, without k<=n particular fixed points.
%F A374419 T(n,k) = T(n,k-1) - T(n-1,k-1) with T(n,0) = A373339(n).
%F A374419 T(n,k) = (1/2) * (Sum_{j=0..k} (-1)^j * binomial(k,j) * (n-j)! + 2^(n-k-1)*(2-n-k)).
%e A374419 Triangle array T(n,k) begins:
%e A374419   n: {k<=n}
%e A374419   0:  {1}
%e A374419   1:  {1,       0}
%e A374419   2:  {1,       0,       0}
%e A374419   3:  {1,       0,       0,       0}
%e A374419   4:  {4,       3,       3,       3,       3}
%e A374419   5:  {36,      32,      29,      26,      23,      20}
%e A374419   6:  {296,     260,     228,     199,     173,     150,     130}
%e A374419   7:  {2360,    2064,    1804,    1576,    1377,    1204,    1054,   924}
%e A374419 T(n,0) = A373339(n) = the number of permutations in S_n without k=0 particular fixed points (i.e., not filtered, so all permutations) with an even number of cycles.
%e A374419 T(n,n) = A216778(n) = the number of permutations in S_n without k=n particular fixed points (i.e., filtered down to just the derangements) with an even number of cycles.
%e A374419 T(4,1<=k<=4) = 3 because S_4 contains 3 permutations with an even number of non-fixed point cycles without k=1,2,3 or 4 particular fixed points, namely the 3 (2,2)-cycles: (12)(34), (13)(24), (14)(23).
%e A374419 T(4,0) = 4 is one more than the above because it includes the permutation without k=0 particular fixed points, i.e., the identity permutation of 4 fixed points.
%t A374419 Table[Table[1/2*(Sum[(-1)^j*Binomial[k, j]*(n - j)!, {j, 0, k}] + 2^(n - k - 1)*(2 - n - k)), {k, 0, n}], {n, 0, 10}]
%Y A374419 Cf. A047920, A062111.
%Y A374419 Cf. A374420 (odd case), A216778 (main diagonal), A373339 (first column).
%K A374419 nonn,tabl
%O A374419 0,11
%A A374419 _Julian Hatfield Iacoponi_, Jul 08 2024