A324354 -id:A324354

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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.

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

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);

# 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);

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.

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

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Feb 17 2019

STATUS

approved

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.

+10
14

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

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

Wikipedia, Permutation

FORMULA

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).

EXAMPLE

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, ...

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 23 2019

STATUS

approved

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