A374420 Triangle T(n, k) for the number of permutations of symmetric group S_n with an odd number of non-fixed point cycles, without k <= n particular fixed points.
0, 0, 0, 1, 1, 1, 5, 4, 3, 2, 20, 15, 11, 8, 6, 84, 64, 49, 38, 30, 24, 424, 340, 276, 227, 189, 159, 135, 2680, 2256, 1916, 1640, 1413, 1224, 1065, 930, 20544, 17864, 15608, 13692, 12052, 10639, 9415, 8350, 7420, 182336, 161792, 143928, 128320, 114628, 102576, 91937, 82522, 74172, 66752
Offset: 0
Examples
Triangle array T(n,k)
n: {k<=n}
0: {0}
1: {0, 0}
2: {1, 1, 1}
3: {5, 4, 3, 2}
4: {20, 15, 11, 8, 6}
5: {84, 64, 49, 38, 30, 24}
6: {424, 340, 276, 227, 189, 159, 135}
7: {2680, 2256, 1916, 1640, 1413, 1224, 1065, 930}
T(n,0) = A373340(n) = the number of permutations in S_n without k=0 particular fixed points (i.e., not filtered, so all permutations) with an odd number of cycles.
T(n,n) = A216779(n) = the number of permutations in S_n without k=n particular fixed points (i.e., filtered down to just the derangements) with an odd number of cycles.
T(2,k) = 1 because S_2 contains 1 permutation with an odd number of non-fixed point cycles without k=0,1 or 2 particular fixed points, namely the derangement (12).
T(3,2) = 3 because S_3 contains 3 permutations with an odd number of non-fixed point cycles without k=2 particular fixed points: say, without fixed points (1) and (2), namely (12)(3), (123), (132).
Crossrefs
Programs
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Mathematica
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}]
Formula
T(n,k) = T(n,k-1) - T(n-1,k-1) with T(n,0) = A373340(n).
T(n,k) = (1/2)*(Sum_{j=0..k} (-1)^j * binomial(k,j) * (n-j)! - 2^(n-k-1)*(2-n-k)).
T(n,k) = (A047920(n, k) + 2^(n-k-1)*(n+k-2))/2. - Peter Luschny, Jul 28 2024