A161133 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k odd fixed points (0 <= k <= ceiling(n/2)).
1, 0, 1, 1, 1, 3, 2, 1, 14, 8, 2, 64, 42, 12, 2, 426, 234, 54, 6, 2790, 1704, 468, 72, 6, 24024, 12864, 3024, 384, 24, 205056, 120120, 32160, 5040, 480, 24, 2170680, 1145400, 272400, 37200, 3000, 120, 22852200, 13024080, 3436200, 544800, 55800
Offset: 0
Examples
T(3,0)=3 because we have 312, 231, 321; T(3,2)=1 because we have 123.
Triangle starts:
1;
0, 1;
1, 1;
3, 2, 1;
14, 8, 2;
64, 42, 12, 2;
426, 234, 54, 6;
Links
- Indranil Ghosh, Rows 0..100, flattened
Programs
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Maple
T := proc (n, k) options operator, arrow: binomial(ceil((1/2)*n), k)*add((-1)^j*binomial(ceil((1/2)*n)-k, j)*factorial(n-k-j), j = 0 .. ceil((1/2)*n)-k) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form
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Mathematica
Flatten[Table[Binomial[Ceiling[n/2], k]*Sum[(-1)^j*(n - k - j)!*Binomial[Ceiling[n/2] - k, j], {j, 0, Ceiling[n/2] - k}],{n, 0, 11}, {k, 0, Ceiling[n/2]}]] (* Indranil Ghosh, Mar 08 2017 *) -
PARI
tabf(nn) = { for(n=0, nn, for(k = 0, ceil(n/2), print1(binomial(ceil(n/2), k) * sum(j=0, ceil(n/2) - k, (-1)^j*(n - k - j)! * binomial(ceil(n/2) - k, j)),", ");); print();); }; tabf(12); \\ Indranil Ghosh, Mar 08 2017
Formula
T(n,k) = binomial(ceiling(n/2), k)*Sum_{j=0..ceiling(n/2)-k} (-1)^j*(n-k-j)!*binomial(ceiling(n/2)-k, j).
Comments