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A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A333706 #36 Feb 09 2023 14:55:10
%S A333706 1,0,1,0,0,2,0,0,4,6,0,2,16,20,24,0,14,44,80,108,120,0,90,200,384,544,
%T A333706 672,720,0,646,1288,2240,3264,4128,4800,5040,0,5242,9512,15424,23040,
%U A333706 28992,34752,38880,40320,0,47622,78652,123456,176832,231936,280512,323520,352800,362880
%N A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A333706 T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.
%H A333706 Alois P. Heinz, <a href="/A333706/b333706.txt">Rows n = 0..20, flattened</a>
%H A333706 Roberto Tauraso, <a href="http://www.emis.de/journals/INTEGERS/papers/g11/g11.Abstract.html">The Dinner Table Problem: The Rectangular Case</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.
%H A333706 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A333706 Triangle T(n,k) begins:
%e A333706   1;
%e A333706   0,    1;
%e A333706   0,    0,    2;
%e A333706   0,    0,    4,     6;
%e A333706   0,    2,   16,    20,    24;
%e A333706   0,   14,   44,    80,   108,   120;
%e A333706   0,   90,  200,   384,   544,   672,   720;
%e A333706   0,  646, 1288,  2240,  3264,  4128,  4800,  5040;
%e A333706   0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320;
%e A333706   ...
%Y A333706 Columns k=0-10 (for n>=k) give: A000007, A002464, A110128, A117574, A189255, A189256, A189271, A360384, A360386, A360462, A360463.
%Y A333706 Main diagonal gives A000142.
%Y A333706 T(2n,n) gives A189849.
%Y A333706 T(n+1,n) gives 4*A138772(n).
%Y A333706 T(n+2,n) gives 16*A333804(n).
%Y A333706 Cf. A000170 (condition is satisfied for all k), A248686 (p(i) at distance k are sorted).
%K A333706 nonn,tabl
%O A333706 0,6
%A A333706 _Alois P. Heinz_, Apr 02 2020