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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

%I

%S 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,

%T 14,34,117,1854,1,1,2,6,24,53,176,792,14833,1,1,2,6,24,78,265,1106,

%U 6205,133496,1,1,2,6,24,120,362,1554,8241,55005,1334961

%N 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.

%H Alois P. Heinz, <a href="/A306512/b306512.txt">Antidiagonals n = 0..45, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

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

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

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

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

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

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 2, 2, 2, 2, 2, ...

%e 2, 2, 3, 6, 6, 6, 6, 6, ...

%e 9, 5, 9, 14, 24, 24, 24, 24, ...

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

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

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

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

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

%p end:

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

%p # second Maple program:

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

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

%p end:

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

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

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

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

%Y A(2n,n) gives A306535.

%Y Cf. A306506.

%K nonn,tabl

%O 0,10

%A _Alois P. Heinz_, Feb 20 2019

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Last modified February 23 07:28 EST 2020. Contains 332159 sequences. (Running on oeis4.)