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A306234
Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n] divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
18
1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 15, 13, 5, 1, 1, 7, 28, 67, 76, 67, 28, 7, 1, 1, 9, 49, 179, 411, 455, 411, 179, 49, 9, 1, 1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 23633, 10757, 3181, 679, 109, 13, 1
OFFSET
1,6
LINKS
Wikipedia, Permutation
FORMULA
T(n,k) = T(n,-k).
T(n,k) = -1/|k|! * Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.
T(n,k) = (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).
T(n+1,n) = 1.
T(n,k) = A306461(n,k) / |k|!.
Sum_{k=1-n..n-1} |k|! * T(n,k) = A306455(n).
EXAMPLE
Triangle T(n,k) begins:
: 1 ;
: 1, 1, 1 ;
: 1, 3, 4, 3, 1 ;
: 1, 5, 13, 15, 13, 5, 1 ;
: 1, 7, 28, 67, 76, 67, 28, 7, 1 ;
: 1, 9, 49, 179, 411, 455, 411, 179, 49, 9, 1 ;
: 1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1 ;
MAPLE
b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, {})):
seq(T(n), n=1..8);
# Alternative:
T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n)/abs(k)!:
seq(seq(T(n, k), k=1-n..n-1), n=1..9);
MATHEMATICA
T[n_, k_] := (-1/Abs[k]!) Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
Table[T[n, k], {n, 1, 9}, {k, 1-n, n-1}] // Flatten (* Jean-François Alcover, Feb 15 2021 *)
CROSSREFS
Columns k=0-10 give (offsets may differ): A002467, A180191, A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
Row sums give A306525.
T(n+1,n) gives A000012.
T(n+2,n) gives A005408.
T(n+2,n-1) gives A056107.
T(2n,n) gives A324361.
Sequence in context: A111028 A201162 A096646 * A290057 A384117 A384118
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Feb 17 2019
STATUS
approved