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A306512 Number A(n,k) of permutations p of [n] having no index i with |p(i)-i| = k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 6
1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 9, 1, 1, 2, 3, 5, 44, 1, 1, 2, 6, 9, 21, 265, 1, 1, 2, 6, 14, 34, 117, 1854, 1, 1, 2, 6, 24, 53, 176, 792, 14833, 1, 1, 2, 6, 24, 78, 265, 1106, 6205, 133496, 1, 1, 2, 6, 24, 120, 362, 1554, 8241, 55005, 1334961 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,10

LINKS

Alois P. Heinz, Antidiagonals n = 0..45, flattened

Wikipedia, Permutation

FORMULA

A(n,k) = n! - A306506(n,k).

A(n,n+i) = n! for i >= 0.

EXAMPLE

A(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.

A(4,1) = 5: 1234, 1432, 3214, 3412, 4231.

A(4,2) = 9: 1234, 1243, 1324, 2134, 2143, 2341, 4123, 4231, 4321.

Square array A(n,k) begins:

     1,   1,    1,    1,    1,    1,    1,    1, ...

     0,   1,    1,    1,    1,    1,    1,    1, ...

     1,   1,    2,    2,    2,    2,    2,    2, ...

     2,   2,    3,    6,    6,    6,    6,    6, ...

     9,   5,    9,   14,   24,   24,   24,   24, ...

    44,  21,   34,   53,   78,  120,  120,  120, ...

   265, 117,  176,  265,  362,  504,  720,  720, ...

  1854, 792, 1106, 1554, 2119, 2790, 3720, 5040, ...

MAPLE

A:= proc(n, k) option remember; `if`(k>=n, n!, LinearAlgebra[

      Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)=k, 0, 1))))

    end:

seq(seq(A(n, d-n), n=0..d), d=0..12);

# second Maple program:

b:= proc(s, k) option remember; (n-> `if`(n=0, 1, add(

      `if`(abs(i-n)=k, 0, b(s minus {i}, k)), i=s)))(nops(s))

    end:

A:= (n, k)-> `if`(k>=n, n!, b({$1..n}, k)):

seq(seq(A(n, d-n), n=0..d), d=0..12);

CROSSREFS

Columns k=0-3 give: A000166, A078480, A306523, A324365.

A(n+2j,n+j) (j=0..5) give: A000142, A001564, A001688, A023043, A023045, A023047.

A(2n,n) gives A306535.

Cf. A306506.

Sequence in context: A114551 A191620 A214751 * A239397 A307884 A329221

Adjacent sequences:  A306509 A306510 A306511 * A306513 A306514 A306515

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 20 2019

STATUS

approved

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Last modified January 24 19:19 EST 2020. Contains 331211 sequences. (Running on oeis4.)