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A306543 Number T(n,k) of permutations p of [n] such that |p(j)-j| >= k (for all j in [n]); triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

%I A306543 #37 Mar 26 2021 19:28:16
%S A306543 1,1,2,1,6,2,24,9,1,120,44,4,720,265,29,1,5040,1854,206,8,40320,14833,
%T A306543 1708,112,1,362880,133496,15702,1168,16,3628800,1334961,159737,13365,
%U A306543 436,1,39916800,14684570,1780696,159414,6984,32,479001600,176214841,21599745,2036488,114124,1708,1
%N A306543 Number T(n,k) of permutations p of [n] such that |p(j)-j| >= k (for all j in [n]); triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.
%H A306543 Alois P. Heinz, <a href="/A306543/b306543.txt">Rows n = 0..22, flattened</a>
%H A306543 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A306543 T(n,k) = Sum_{j=k..floor(n/2)} A299789(n,j) for n > 0.
%e A306543 Triangle T(n,k) begins:
%e A306543           1;
%e A306543           1;
%e A306543           2,         1;
%e A306543           6,         2;
%e A306543          24,         9,        1;
%e A306543         120,        44,        4;
%e A306543         720,       265,       29,       1;
%e A306543        5040,      1854,      206,       8;
%e A306543       40320,     14833,     1708,     112,      1;
%e A306543      362880,    133496,    15702,    1168,     16;
%e A306543     3628800,   1334961,   159737,   13365,    436,    1;
%e A306543    39916800,  14684570,  1780696,  159414,   6984,   32;
%e A306543   479001600, 176214841, 21599745, 2036488, 114124, 1708, 1;
%e A306543   ...
%p A306543 T:= proc(n, k) option remember; `if`(n=0, 1, LinearAlgebra[
%p A306543       Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))
%p A306543     end:
%p A306543 seq(seq(T(n, k), k=0..floor(n/2)), n=0..12);
%t A306543 T[n_, k_] := T[n, k] = If[n==0, 1, Permanent[Table[
%t A306543      If[Abs[i-j] >= k, 1, 0], {i, n}, {j, n}]]];
%t A306543 Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Mar 26 2021, after _Alois P. Heinz_ *)
%Y A306543 Columns k=0-6 give (offsets may differ): A000142, A000166, A001883, A075851, A075852, A183242, A183243.
%Y A306543 T(2n,n) gives A000012.
%Y A306543 T(2n+1,n) gives A000079.
%Y A306543 T(2n+2,n) gives A183245 for n > 0.
%Y A306543 T(2n+3,n) gives A183246 for n > 0.
%Y A306543 T(2n+4,n) gives A183247 for n > 0.
%Y A306543 Cf. A183244, A299789.
%K A306543 nonn,tabf
%O A306543 0,3
%A A306543 _Alois P. Heinz_, Feb 22 2019