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 *)
A324362
Total number of occurrences of k in the (signed) displacement sets of all permutations of [n+k] divided by k!; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
0, 0, 1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 5, 13, 15, 0, 1, 7, 28, 67, 76, 0, 1, 9, 49, 179, 411, 455, 0, 1, 11, 76, 375, 1306, 2921, 3186, 0, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 0, 1, 15, 148, 1115, 6576, 29843, 98932, 214551, 229384, 0, 1, 17, 193, 1707, 12151, 69299, 307833, 1006007, 2160343, 2293839
Offset: 0
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, 13, ...
4, 13, 28, 49, 76, 109, 148, ...
15, 67, 179, 375, 679, 1115, 1707, ...
76, 411, 1306, 3181, 6576, 12151, 20686, ...
455, 2921, 10757, 29843, 69299, 142205, 266321, ...
Columns k=0-10 give:
A002467,
A180191(n+1),
A324352,
A324353,
A324354,
A324355,
A324356,
A324357,
A324358,
A324359,
A324360.
-
A:= (n, k)-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!:
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
m = 10;
col[k_] := col[k] = CoefficientList[(1-Exp[-x])/(1-x)^(k+1)+O[x]^(m+1), x]* Range[0, m]!;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 03 2021 *)
A386965
Number of permutations p of [2*n] such that there is at least one index i in [2*n-1] with p(i+1) = n + p(i).
Original entry on oeis.org
1, 10, 294, 16296, 1458120, 191751120, 34807535280, 8337722440320, 2547572372311680, 966944845408147200, 446304490431888211200, 246166572372916851532800, 159902551429370021259187200, 120818209587660157360960972800, 105060730670227917425027835648000
Offset: 1
The 10 permutations corresponding to a(2) are 1243, 1324, 1342, 2134, 2413, 2431, 3124, 3241, 4132, 4213.
-
a[n_] := Sum[(-1)^(k+1) Binomial[n, k] (2 n - k)!, {k, n}]; Array[a, 15]
Showing 1-3 of 3 results.
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