login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A306461 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n]; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows. 6
1, 1, 1, 1, 2, 3, 4, 3, 2, 6, 10, 13, 15, 13, 10, 6, 24, 42, 56, 67, 76, 67, 56, 42, 24, 120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120, 720, 1320, 1824, 2250, 2612, 2921, 3186, 2921, 2612, 2250, 1824, 1320, 720, 5040, 9360, 13080, 16296, 19086, 21514, 23633, 25487, 23633, 21514, 19086, 16296, 13080, 9360, 5040 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

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

Wikipedia, Permutation

FORMULA

T(n,k) = T(n,-k).

T(n,k) = - Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.

T(n,k) = |k|! * (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).

Sum_{k=1-n..n-1} T(n,k) = A306455(n).

T(n,k) = |k|! * A306234(n,k).

EXAMPLE

The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement sets {p(i)-i, i=1..3}: {0}, {-1,0,1}, {-1,0,1}, {-2,1}, {-1,2}, {-2,0,2}, respectively. Numbers -2 and 2 occur twice, -1 and 1 occur thrice, and 0 occurs four times. So row n=3 is [2, 3, 4, 3, 2].

Triangle T(n,k) begins:

  :                             1                           ;

  :                        1,   1,   1                      ;

  :                   2,   3,   4,   3,   2                 ;

  :              6,  10,  13,  15,  13,  10,   6            ;

  :        24,  42,  56,  67,  76,  67,  56,  42,  24       ;

  :  120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120  ;

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), 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):

seq(seq(T(n, k), k=1-n..n-1), n=1..9);

CROSSREFS

Columns k=0-1 give: A002467, A180191.

Row sums give A306455.

T(n+1,n) gives A000142.

T(n+2,n) gives A007680.

Cf. A000142, A061018 (left half of this triangle), A306234, A306506, A324225.

Sequence in context: A251102 A217287 A111880 * A101497 A274007 A065870

Adjacent sequences:  A306458 A306459 A306460 * A306462 A306463 A306464

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Feb 17 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 29 01:54 EST 2020. Contains 331328 sequences. (Running on oeis4.)