A233440 Triangle read by rows: T(n, k) = n*binomial(n, k)*A000757(k), 0 <= k <= n.
0, 1, 0, 2, 0, 0, 3, 0, 0, 3, 4, 0, 0, 16, 4, 5, 0, 0, 50, 25, 40, 6, 0, 0, 120, 90, 288, 216, 7, 0, 0, 245, 245, 1176, 1764, 1603, 8, 0, 0, 448, 560, 3584, 8064, 14656, 13000, 9, 0, 0, 756, 1134, 9072, 27216, 74196, 131625, 118872, 10, 0, 0, 1200, 2100, 20160, 75600, 274800, 731250, 1320800, 1202880
Offset: 0
Examples
For n = 4 and k = 4, T(4, 4) = 4 because all the permutations of 4 symbols that 4-commute with permutation (1, 2, 3, 4) are (1, 3), (2, 4), (1, 2)(3, 4) and (1, 4)(2, 3).
Links
- Luis Manuel Rivera Martínez, Rows n = 0..30 of triangle, flattened
- R. Moreno and L. M. Rivera, Blocks in cycles and k-commuting permutations, arXiv:1306.5708 [math.CO], 2013-2014.
- Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Programs
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Mathematica
T[n_, k_] := n Binomial[n, k] ((-1)^k+Sum[(-1)^j k!/(k-j)/j!, {j, 0, k-1}]); Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)
Formula
T(n,k) = n*C(n,k)*A000757(k), 0 <= k <= n.
Bivariate e.g.f.: G(z, u) = z*exp(z*(1-u))*(u/(1-z*u)+(1-log(1-z*u))*(1-u)).
T(n, 0) = A001477(n), n>=0;
T(n, 1) = A000004(n), n>=1;
T(n, 2) = A000004(n), n>=2;
T(n, 3) = A004320(n-2), n>=3;
T(n, 4) = A027764(n-1), n>=4;
T(n, 0)+T(n, 3) = n*A050407(n+1), for n>=0. - Luis Manuel Rivera Martínez, Mar 06 2014
Comments