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 *)
A260091
Number of permutations p of [n] with no fixed points and cyclic displacement of elements restricted by seven: p(i)<>i and (i-p(i) mod n <= 7 or p(i)-i mod n <= 7).
Original entry on oeis.org
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 2649865335040, 14570246018686, 80002336342276, 438791546196382, 2404416711392528, 13164695578635648, 72030936564665508, 393911127182051942
Offset: 0
-
a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](Matrix(n, (i, j)->
`if`(i<>j and (i-j mod n<=7 or j-i mod n<=7), 1, 0)))):
seq(a(n), n=0..16);
-
a[n_] := If[n == 0, 1, Permanent[Table[If[i != j && (Mod[i - j, n] <= 7 || Mod[j - i, n] <= 7), 1, 0], {i, 1, n}, {j, 1, n}]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 16}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)
A321053
Number of permutations of [n] with no fixed points where the maximal displacement of an element equals seven.
Original entry on oeis.org
0, 3973, 46662, 359724, 2270840, 12994207, 71401928, 389384029, 2162806868, 12393816965, 72063175820, 420431229772, 2443816138204, 14098735154983, 80635213447724, 457661900843292, 2583741730021382, 14552232847553984, 81904772313605164, 461011600699499344
Offset: 7
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
Showing 1-4 of 4 results.
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