A188497 -id:A188497 - cp's OEIS frontend

A188493 a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).

0, 0, 2, 6, 14, 31, 104, 344, 1084, 3236, 9784, 29964, 92241, 282780, 865064, 2646292, 8102454, 24813838, 75982346, 232630527, 712230076, 2180675264, 6676819512, 20443032008, 62591840320, 191641545768, 586762729889, 1796535598952, 5500587026592

Offset: 0

N. J. A. Sloane, Apr 01 2011

nonn

For n >= 2, a(n) is the number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(j) <= 2+j for j = 1,2, and p(j) >= j-2 for j = 4,5.
For n >= 2, a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of zeros in the (1,4) and (2,5)-entries), ones on its three subdiagonals (with the exception of zeros in the (4,1) and (5,2)-entries), and is zero elsewhere.
This is row 7 of Kløve's Table 3.

Nathaniel Johnston, Table of n, a(n) for n = 0..124
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

with (LinearAlgebra):
A188493:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[4, 1] and [i, j]<>[5, 2] and [i, j]<>[1, 4] and [i, j]<>[2, 5], 1, 0)))):
seq (A188493(n), n=0..20);

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {4, 1} && {i, j} != {5, 2} && {i, j} != {1, 4} && {i, j} != {2, 5}, 1, 0], {i, 1, n}, {j, 1, n}] ]; a[1] = 0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)

G.f.: -(x^10+2*x^9+2*x^7 +4*x^6-2*x^5-6*x^4 -9*x^3-2*x^2+2*x+2) *x^2 / (x^14 +2*x^13+2*x^11 +4*x^10-2*x^9-10*x^8 -16*x^7-2*x^6+8*x^5 +10*x^4 +2*x^2 +2*x-1). - Alois P. Heinz, Apr 08 2011

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 08 2011

A002526 Number of permutations of length n within distance 3 of a fixed permutation.

Original entry on oeis.org

1, 1, 2, 6, 24, 78, 230, 675, 2069, 6404, 19708, 60216, 183988, 563172, 1725349, 5284109, 16177694, 49526506, 151635752, 464286962, 1421566698, 4352505527, 13326304313, 40802053896, 124926806216, 382497958000, 1171122069784, 3585709284968, 10978628154457

Offset: 0

N. J. A. Sloane

For positive n, a(n) equals the permanent of the n X n matrix with 1's along the seven central diagonals, and 0's everywhere else. - John M. Campbell, Jul 09 2011

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. H. Hardin, Table of n, a(n) for n=0..400, Jul 11 2010
V. Baltic, On the number of certain types of strongly restricted permutations, Appl. An. Disc. Math. 4 (2010), 119-135; DOI:10.2298/AADM1000008B.
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008. (Table 3, top row).
O. Krafft and M. Schaefer, On the number of permutations within a given distance, Fib. Quart. 40 (5) (2002) 429-434.
R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).

The 14 sequences in Kløve's Table 3 are A002526, A002527, A002529, A188379, A188491, A188492, A188493, A188494, A002528, A188495, A188496, A188497, A188498, A002526.
Cf. A002524.
Column k=3 of A306209.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14) )); // G. C. Greubel, Jan 22 2022

CoefficientList[Series[(1-x-2x^2-2x^4+x^7+x^8)/(1-2x-2x^2-10x^4-8x^5+ 2x^6+ 16x^7+10x^8+2x^9-4x^10-2x^11-2x^13-x^14),{x,0,50}],x] (* or *) LinearRecurrence[{2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1},{1,1,2,6,24,78, 230, 675,2069,6404,19708,60216,183988,563172},51] (* Harvey P. Dale, Jun 22 2011 *)

Vec((1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8+2*x^9-4*x^10-2*x^11-2*x^13-x^14)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

[( (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Jan 22 2022

G.f.: (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14).

a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=78, a(6)=230, a(7)=675, a(8)=2069, a(9)=6404, a(10)=19708, a(11)=60216, a(12)=183988, a(13)=563172, a(n) = 2*a(n-1) +2*a(n-2) +10*a(n-4) +8*a(n-5) -2*a(n-6) -16*a(n-7) -10*a(n-8) -2*a(n-9) +4*a(n-10) +2*a(n-11) +2*a(n-13) +a(n-14). - Harvey P. Dale, Jun 22 2011

A188495 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, and p(4) >= 2.

Original entry on oeis.org

0, 1, 2, 4, 10, 36, 120, 368, 1089, 3304, 10168, 31312, 95880, 293120, 896824, 2746569, 8411818, 25756220, 78853410, 241421436, 739183568, 2263249600, 6929580817, 21216729488, 64960656448, 198894856144, 608971496032, 1864533223584, 5708777321872

Offset: 0

N. J. A. Sloane, Apr 01 2011

a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of a zero in the (1,4)-entry), ones on its three subdiagonals (with the exception of zeros in the (3,1) and (4,1)-entries), and is zero elsewhere.

This is row 10 of Kløve's Table 3.

Nathaniel Johnston, Table of n, a(n) for n = 0..119
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).

with (LinearAlgebra):
A188495:= n-> `if` (n=0, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[3, 1] and [i, j]<>[4, 1] and [i, j]<>[1, 4], 1, 0)))):
seq (A188495(n), n=0..20);

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {3, 1} && {i, j} != {4, 1} && {i, j} != {1, 4}, 1, 0], {i, 1, n}, {j, 1, n}]]; a[1] = 1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)

concat(0, Vec(-x*(x +1)*(x^6 +x^5 -x^4 -x^3 -x^2 -x +1) / ((x -1)*(x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x +1)) + O(x^100))) \\ Colin Barker, Dec 13 2014

From Nathaniel Johnston, Apr 10 2011: (Start)
a(n) = A188493(n+1) - A188491(n) - A188497(n).
a(n) = A002526(n-1) + A188494(n-1).
(End)
G.f.: -x*(x +1)*(x^6 +x^5 -x^4 -x^3 -x^2 -x +1) / ((x -1)*(x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x +1)). - Colin Barker, Dec 13 2014

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 10 2011

A188498 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, and p(j) >= 2 for j=3,4.

Original entry on oeis.org

0, 1, 2, 3, 8, 30, 102, 308, 905, 2744, 8473, 26112, 79924, 244204, 747160, 2288521, 7009458, 21461803, 65704200, 201162258, 615922714, 1885853660, 5774072225, 17678809840, 54128358209, 165728860112, 507424764216, 1553620027784, 4756831354752

Offset: 0

N. J. A. Sloane, Apr 01 2011

nonn

a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of zeros in the (1,3) and (1,4)-entries), ones on its three subdiagonals (with the exception of zeros in the (3,1) and (4,1)-entries), and is zero elsewhere.

This is row 13 of Kløve's Table 3.

Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

with(LinearAlgebra):
A188498:= n-> `if` (n=0, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[1, 3] and [i, j]<>[1, 4] and [i, j]<>[3, 1] and [i, j]<>[4, 1], 1, 0)))):
seq (A188498(n), n=0..20);

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {1, 3} && {i, j} != {1, 4} && {i, j} != {3, 1} && {i, j} != {4, 1}, 1, 0], {i, 1, n}, {j, 1, n}]]; a[1] = 1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 07 2016, adapted from Maple *)
CoefficientList[Series[-(x^10 + 2 x^9 + x^8 - 2 x^6 - 2 x^5 - 2 x^4 - 3 x^3 + x) / (x^14 + 2 x^13 + 2 x^11 + 4 x^10 - 2 x^9 - 10 x^8 - 16 x^7 - 2 x^6 + 8 x^5 + 10 x^4 + 2 x^2 + 2 x - 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 07 2016 *)

concat(0, Vec(-(x^10+2*x^9+x^8 -2*x^6-2*x^5-2*x^4 -3*x^3+x) / (x^14+2*x^13+2*x^11 +4*x^10-2*x^9-10*x^8 -16*x^7-2*x^6+8*x^5 +10*x^4+2*x^2+2*x-1) + O(x^40))) \\ Michel Marcus, Dec 12 2014

From Nathaniel Johnston, Apr 11 2011: (Start)
a(n) = A188497(n+1) - A188494(n).
a(n) = A002526(n-1) + A002526(n-2).
(End)
G.f.: -(x^10 + 2*x^9 + x^8 - 2*x^6 - 2*x^5 - 2*x^4 - 3*x^3 + x) / (x^14 + 2*x^13 + 2*x^11 + 4*x^10 - 2*x^9 - 10*x^8 - 16*x^7 - 2*x^6 + 8*x^5 + 10*x^4 + 2*x^2 + 2*x - 1).

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 11 2011

cp's OEIS Frontend

A188493 a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).

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A002526 Number of permutations of length n within distance 3 of a fixed permutation.

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A188495 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, and p(4) >= 2.

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A188498 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, and p(j) >= 2 for j=3,4.

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Programs

Maple

Mathematica

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Formula

Extensions