A259776
Number A(n,k) of permutations p of [n] with no fixed points and displacement of elements restricted by k: 1 <= |p(i)-i| <= k, square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 4, 0, 0, 1, 0, 1, 2, 9, 6, 1, 0, 1, 0, 1, 2, 9, 24, 13, 0, 0, 1, 0, 1, 2, 9, 44, 57, 24, 1, 0, 1, 0, 1, 2, 9, 44, 168, 140, 45, 0, 0, 1, 0, 1, 2, 9, 44, 265, 536, 376, 84, 1, 0
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 0, 2, 2, 2, 2, 2, 2, ...
0, 1, 4, 9, 9, 9, 9, 9, ...
0, 0, 6, 24, 44, 44, 44, 44, ...
0, 1, 13, 57, 168, 265, 265, 265, ...
0, 0, 24, 140, 536, 1280, 1854, 1854, ...
Columns k=0-10 give:
A000007,
A059841,
A033305,
A079997,
A259777,
A259778,
A259779,
A259780,
A259781,
A259782,
A259783.
-
b:= proc(n, s, k) option remember; `if`(n=0, 1, `if`(n+k in s,
b(n-1, (s minus {n+k}) union `if`(n-k>1, {n-k-1}, {}), k),
add(`if`(j=n, 0, b(n-1, (s minus {j}) union
`if`(n-k>1, {n-k-1}, {}), k)), j=s)))
end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, {$max(1, n-k)..n}, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
b[n_, s_, k_] := b[n, s, k] = If[n==0, 1, If[MemberQ[s, n+k], b[n-1, Join[s ~Complement~ {n+k}] ~Union~ If[n-k>1, {n-k-1}, {}], k], Sum[If[j==n, 0, b[n -1, Join[s ~Complement~ {j}] ~Union~ If[n-k>1, {n-k-1}, {}], k]], {j, s}]] ];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Range[Max[1, n-k], n], k]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 29 2017, translated from Maple *)
A260094
Number of permutations p of [n] with no fixed points and cyclic displacement of elements restricted by five: p(i)<>i and (i-p(i) mod n <= 5 or p(i)-i mod n <= 5).
Original entry on oeis.org
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 59245120, 238282730, 956135652, 3828509472, 15296722436, 60990443730, 243596762752, 975165838970, 3913571754304, 15742403448024, 63428117376852, 255662480209770, 1031080275942464, 4161127398011040
Offset: 0
-
a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](Matrix(n, (i, j)->
`if`(i<>j and (i-j mod n<=5 or j-i mod n<=5), 1, 0)))):
seq(a(n), n=0..15);
-
a[n_] := If[n == 0, 1, Permanent[Table[If[i != j && (Mod[i - j, n] <= 5 || Mod[j - i, n] <= 5), 1, 0], {i, 1, n}, {j, 1, n}]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)
A321051
Number of permutations of [n] with no fixed points where the maximal displacement of an element equals five.
Original entry on oeis.org
0, 97, 744, 3628, 15038, 59963, 245386, 1054116, 4585666, 19816317, 84377528, 354143757, 1472404694, 6105113905, 25314316974, 105009933736, 435538239600, 1804712603943, 7467462109856, 30859243303779, 127410491972804, 525761755401512, 2168906840165128
Offset: 5
A321052
Number of permutations of [n] with no fixed points where the maximal displacement of an element equals six.
Original entry on oeis.org
0, 574, 5571, 34948, 181193, 870934, 4113244, 19845700, 99472963, 505871096, 2572439079, 12975021278, 64715221044, 319501249574, 1566516774344, 7660714576044, 37445731303872, 183081927465284, 895275467752721, 4376669424005308, 21379883128454905
Offset: 6
Showing 1-4 of 4 results.
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