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 *)
A260115
Number of permutations p of [n] with no fixed points and cyclic displacement of elements restricted by eight: p(i)<>i and (i-p(i) mod n <= 8 or p(i)-i mod n <= 8).
Original entry on oeis.org
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 817154768973824, 5095853023109484, 31742020729513344, 197541094675490640, 1228455950686697872, 7634711586761705092
Offset: 0
-
a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](Matrix(n, (i, j)->
`if`(i<>j and (i-j mod n<=8 or j-i mod n<=8), 1, 0)))):
seq(a(n), n=0..18);
-
a[n_] := If[n == 0, 1, Permanent[Table[If[i != j && (Mod[i - j, n] <= 8 || Mod[j - i, n] <= 8), 1, 0], {i, 1, n}, {j, 1, n}]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 18}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)
A321054
Number of permutations of [n] with no fixed points where the maximal displacement of an element equals eight.
Original entry on oeis.org
0, 31520, 434127, 3979676, 29836261, 200251036, 1269799244, 7866814026, 48627012403, 305484315674, 1970229533360, 12885942393622, 84719993481585, 556551975117208, 3639388789709545, 23644099935731588, 152553455368658432, 978410935801325180, 6248373677853677799
Offset: 8
A321055
Number of permutations of [n] with no fixed points where the maximal displacement of an element equals nine.
Original entry on oeis.org
0, 281825, 4451724, 47355884, 412882090, 3202361483, 23212224128, 162423570785, 1121100331330, 7743988554848, 54274875428464, 389110611485999, 2826724445254744, 20662487594362983, 151199827593760314, 1103728723985787116, 8020129818824497034
Offset: 9
Showing 1-4 of 4 results.
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