OFFSET
1,3
COMMENTS
T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n].
LINKS
Alois P. Heinz, Rows n = 1..100, flattened
Nadir Samos Sáenz de Buruaga, Rafał Bistroń, Marcin Rudziński, Rodrigo Miguel Chinita Pereira, Karol Życzkowski, and Pedro Ribeiro, Fidelity decay and error accumulation in quantum volume circuits, arXiv:2404.11444 [quant-ph], 2024. See p. 18.
Wikipedia, Permutation
Wikipedia, Permutation matrix
FORMULA
EXAMPLE
The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices
[1 ] [1 ] [ 1 ] [ 1 ] [ 1] [ 1]
[ 1 ] [ 1] [1 ] [ 1] [1 ] [ 1 ]
[ 1] [ 1 ] [ 1] [1 ] [ 1 ] [1 ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].
Triangle T(n,k) begins:
: 1 ;
: 1, 2, 1 ;
: 2, 4, 6, 4, 2 ;
: 6, 12, 18, 24, 18, 12, 6 ;
: 24, 48, 72, 96, 120, 96, 72, 48, 24 ;
: 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120 ;
MAPLE
b:= proc(s, c) option remember; (n-> `if`(n=0, c,
add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!, 0))(abs(k)):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
MATHEMATICA
T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!, 0]];
Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 25 2021, after 3rd Maple program *)
CROSSREFS
Columns k=0-6 give (offsets may differ): A000142, A001563, A062119, A052571, A052520, A282822, A052521.
Row sums give A001563.
T(n+1,n) gives A000142.
T(n+1,n-1) gives A052849.
T(n+1,n-2) gives A052560 for n>1.
Cf. A152883 (right half of this triangle without center column), A162608 (left half of this triangle), A306461, A324224.
Cf. A001710.
KEYWORD
AUTHOR
Alois P. Heinz, Feb 18 2019
STATUS
approved