A259783 -id:A259783 - cp's OEIS frontend

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.

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

Alois P. Heinz, Jul 05 2015

Conjecture: Column k > 0 has a linear recurrence (with constant coefficients) of order = A005317(k) = (2^k + C(2*k,k))/2. - Vaclav Kotesovec, Jul 07 2015
From Vaclav Kotesovec, Jul 07 2015: (Start) For k > 1, A(n,k) ~ c(k) * d(k)^n
k  c(k)                                  d(k)
2  0.2840509026895102746628049030651...  1.8832035059135258641689474653620...
3  0.1678494211968692989590951622212...  2.6304414743928951523517253855770...
4  0.0973070675347403976445165510589...  3.3758288741377846847522960161445...
5  0.0552389982575367440330445172521...  4.1183824671958029895499633437571...
6  0.0309726120341077011398575643793...  4.8588208495640240252838055706997...
7  0.0172064353582683268003622374813...  5.5979905586951369718393573797927...
8  0.0094902135663231445267663712259...  6.3363450921766600853069060904417...
9  0.00520430877801650454166967632...    7.0741444217884608367707985...
10 0.0028405987031922...                 7.811548995086...
(End)

			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, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0-10 give: A000007, A059841, A033305, A079997, A259777, A259778, A259779, A259780, A259781, A259782, A259783.
Main diagonal gives: A000166.
Cf. A259784.

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 *)

A(n,k) = Sum_{j=0..k} A259784(n,j).

A260216 Number of permutations p of [n] with no fixed points and cyclic displacement of elements restricted by ten: p(i)<>i and (i-p(i) mod n <= 10 or p(i)-i mod n <= 10).

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540, 145060238642780180480, 1118480911876659396600

Offset: 0

Alois P. Heinz, Jul 19 2015

nonn

a(n) = A000166(n) for n <= 21.

Cf. A000166, A260074, A260081, A260092, A260094, A260111, A260091, A260115, A257953.

Cf. A259783.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](Matrix(n, (i, j)->
        `if`(i<>j and (i-j mod n<=10 or j-i mod n<=10), 1, 0)))):
seq(a(n), n=0..22);

a[n_] := If[n == 0, 1, Permanent[Table[If[i != j && (Mod[i - j, n] <= 10 || Mod[j - i, n] <= 10), 1, 0], {i, 1, n}, {j, 1, n}]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 22}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)

A321056 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals ten.

Original entry on oeis.org

0, 2803418, 49942371, 605109972, 6030258697, 53315654282, 437476633564, 3434708277908, 26357552759183, 200667318966580, 1531612782695209, 11845709562106324, 93455676821980373, 746513630305265464, 6002963117390580668, 48383934184947635708, 389641866674965457439

Offset: 10

Alois P. Heinz, Oct 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..175

Column k=10 of A259784.

Cf. A259782, A259783.

a(n) = A259783(n) - A259782(n).

cp's OEIS Frontend

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.

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A260216 Number of permutations p of [n] with no fixed points and cyclic displacement of elements restricted by ten: p(i)<>i and (i-p(i) mod n <= 10 or p(i)-i mod n <= 10).

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Author

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Comments

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Programs

Maple

Mathematica

A321056 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals ten.

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Author

Keywords

Links

Crossrefs

Formula