A324359 -id:A324359 - cp's OEIS frontend

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.

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

Alois P. Heinz, Feb 17 2019

			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  ;

Alois P. Heinz, Rows n = 1..142, flattened
Wikipedia, Permutation

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.
Cf. A000142, A306455, A306461, A324224, A324362.

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 *)

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).

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

Alois P. Heinz, Feb 23 2019

			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, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A002467, A180191(n+1), A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
Rows n=0-3 give: A000004, A000012, A005408, A056107(k+1).
Main diagonal gives A324361.
Cf. A306234.

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 *)

E.g.f. of column k: (1-exp(-x))/(1-x)^(k+1).
A(n,k) = -1/k! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+k-j)!.
A(n,k) = A306234(n+k,k).

cp's OEIS Frontend

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.

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