A002526 Number of permutations of length n within distance 3 of a fixed permutation.
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
References
- 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).
Links
- 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).
Crossrefs
Programs
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Magma
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 -
Mathematica
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 *) -
PARI
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
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Sage
[( (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
Formula
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
Comments