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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. 15

%I

%S 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 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 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 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 A(n,k) counts permutations p of [n] such that |p(j)-j| <= k for all j in [n].

%H Alois P. Heinz, <a href="/A306209/b306209.txt">Antidiagonals n = 0..36, flattened</a>

%H 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 Torleiv Kløve, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r104">Generating functions for the number of permutations with limited displacement</a>, Electron. J. Combin., 16 (2009), #R104.

%F A(n,k) = Sum_{j=0..k} A130152(n,j) for n > 0, A(0,k) = 1.

%e A(4,1) = 5: 1234, 1243, 1324, 2134, 2143.

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

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

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

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

%e 1, 3, 6, 6, 6, 6, 6, 6, 6, ...

%e 1, 5, 14, 24, 24, 24, 24, 24, 24, ...

%e 1, 8, 31, 78, 120, 120, 120, 120, 120, ...

%e 1, 13, 73, 230, 504, 720, 720, 720, 720, ...

%e 1, 21, 172, 675, 1902, 3720, 5040, 5040, 5040, ...

%e 1, 34, 400, 2069, 6902, 17304, 30960, 40320, 40320, ...

%Y Columns k=0..10 give: A000012, A000045(n+1), A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659.

%Y Rows n=1-2 give: A000012, A040000.

%Y Main diagonal gives A000142.

%Y A(2n,n) gives A048163(n+1).

%Y A(2n+1,n) gives A092552(n+1).

%Y A(n,floor(n/2)) gives A306267.

%Y A(n+2,n) gives A001564.

%Y Cf. A130152.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Jan 29 2019

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Last modified December 12 12:50 EST 2019. Contains 329958 sequences. (Running on oeis4.)