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.
Original entry on oeis.org
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
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 ;
Columns k=0-10 give (offsets may differ):
A002467,
A180191,
A324352,
A324353,
A324354,
A324355,
A324356,
A324357,
A324358,
A324359,
A324360.
-
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);
# second Maple program:
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);
-
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 *)
A324225
Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
Original entry on oeis.org
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, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040
Offset: 1
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 ;
- 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
-
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
-
T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)
A306495
Expansion of e.g.f. (2-exp(-x))*exp(x)/(x-1)^2.
Original entry on oeis.org
1, 4, 16, 74, 402, 2542, 18446, 151482, 1390738, 14126582, 157365222, 1908110866, 25022451482, 352918443438, 5327630246542, 85716034274282, 1464281837606946, 26470821156031462, 504879319309407158, 10132393298394712002, 213441590598213760042
Offset: 0
-
egf:= (2-exp(-x))*exp(x)/(x-1)^2:
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..23);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 4^n,
(2*n+1)*a(n-1)-(n+2)*(n-1)*a(n-2)+(n-1)*(n-2)*a(n-3))
end:
seq(a(n), n=0..23);
Showing 1-3 of 3 results.
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