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A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A306209 #36 Oct 18 2021 11:13:24
%S A306209 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,6,5,1,1,1,2,6,14,8,1,1,1,2,6,24,
%T A306209 31,13,1,1,1,2,6,24,78,73,21,1,1,1,2,6,24,120,230,172,34,1,1,1,2,6,24,
%U A306209 120,504,675,400,55,1,1,1,2,6,24,120,720,1902,2069,932,89,1,1,1,2,6,24,120,720,3720,6902,6404,2177,144,1
%N A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A306209 A(n,k) counts permutations p of [n] such that |p(j)-j| <= k for all j in [n].
%H A306209 Alois P. Heinz, <a href="/A306209/b306209.txt">Antidiagonals n = 0..36, flattened</a>
%H A306209 Torleiv Kløve, <a href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2008-376.pdf">Spheres of Permutations under the Infinity Norm - Permutations with limited displacement</a>, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
%H A306209 Torleiv Kløve, <a href="https://doi.org/10.37236/193">Generating functions for the number of permutations with limited displacement</a>, Electron. J. Combin., 16 (2009), #R104.
%F A306209 A(n,k) = Sum_{j=0..k} A130152(n,j) for n > 0, A(0,k) = 1.
%e A306209 A(4,1) = 5: 1234, 1243, 1324, 2134, 2143.
%e A306209 A(5,2) = 31: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14253, 14325, 14523, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 24153, 31245, 31254, 31425, 31524, 32145, 32154, 34125.
%e A306209 Square array A(n,k) begins:
%e A306209   1,  1,   1,    1,    1,     1,     1,     1,     1, ...
%e A306209   1,  1,   1,    1,    1,     1,     1,     1,     1, ...
%e A306209   1,  2,   2,    2,    2,     2,     2,     2,     2, ...
%e A306209   1,  3,   6,    6,    6,     6,     6,     6,     6, ...
%e A306209   1,  5,  14,   24,   24,    24,    24,    24,    24, ...
%e A306209   1,  8,  31,   78,  120,   120,   120,   120,   120, ...
%e A306209   1, 13,  73,  230,  504,   720,   720,   720,   720, ...
%e A306209   1, 21, 172,  675, 1902,  3720,  5040,  5040,  5040, ...
%e A306209   1, 34, 400, 2069, 6902, 17304, 30960, 40320, 40320, ...
%t A306209 A[0, _] = 1;
%t A306209 A[n_ /; n > 0, k_] := A[n, k] = Permanent[Table[If[Abs[i - j] <= k, 1, 0], {i, 1, n}, {j, 1, n}]];
%t A306209 Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1 }] // Flatten (* _Jean-François Alcover_, Oct 18 2021, after _Alois P. Heinz_ in A130152 *)
%Y A306209 Columns k=0..10 give: A000012, A000045(n+1), A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659.
%Y A306209 Rows n=1-2 give: A000012, A040000.
%Y A306209 Main diagonal gives A000142.
%Y A306209 A(2n,n) gives A048163(n+1).
%Y A306209 A(2n+1,n) gives A092552(n+1).
%Y A306209 A(n,floor(n/2)) gives A306267.
%Y A306209 A(n+2,n) gives A001564.
%Y A306209 Cf. A130152.
%K A306209 nonn,tabl
%O A306209 0,9
%A A306209 _Alois P. Heinz_, Jan 29 2019