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A002524 Number of permutations of length n within distance 2 of a fixed permutation.
(Formerly M1600 N0626)
86
1, 1, 2, 6, 14, 31, 73, 172, 400, 932, 2177, 5081, 11854, 27662, 64554, 150639, 351521, 820296, 1914208, 4466904, 10423761, 24324417, 56762346, 132458006, 309097942, 721296815, 1683185225, 3927803988, 9165743600, 21388759708, 49911830577, 116471963129 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Torleiv Kløve, Jan 09 2009: (Start)

Let V(d,n) be the number of permutations of length n within distance d of a fixed permutation. For d=1,2,3,4,...,10 these are A000045, A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659. The generating function is a rational function f_d(z)/g_d(z) (see the Kløve report referenced here). For d<=6, deg g_d = 2^{d-1} + binomial(2*d,d)/2 and deg f_d(z) = deg g_d(z)-2d. As a table:

   d deg g_d deg f_d

   1    2       0

   2    5       1

   3   14       8

   4   43      35

   5  142     132

   6  494     482

(End)

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

REFERENCES

V. Baltic, On the number of certain types of strongly restricted permutations, Appl. An. Disc. Math. 4 (2010), 119-135; Doi:10.2298/AADM1000008B; http://pefmath.etf.rs

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. P. Stanley, Enumerative Combinatorics I, Example 4.7.16, p. 253.

LINKS

R. H. Hardin, Table of n, a(n) for n = 0..400, Jul 11 2010

M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25. - From N. J. A. Sloane, Dec 25 2012

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.

Torleiv Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104. - From N. J. A. Sloane, May 04 2011.

Ting Kuo, K-sorted Permutations with Weakly Restricted Displacements, Journal of Computers, 2014. See p. 36.

R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (2,0,2,0,-1).

FORMULA

G.f.: (1-x)/(1-2*x-2*x^3+x^5). - Simon Plouffe in his 1992 dissertation.

MATHEMATICA

CoefficientList[Series[(1 - z)/(z^5 - 2 z^3 - 2 z + 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 24 2011 *)

PROG

(PARI) a(n)=if(n, ([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; -1, 0, 2, 0, 2]^n*[1; 1; 2; 6; 14])[1, 1], 1) \\ Charles R Greathouse IV, Jul 28 2015

CROSSREFS

Sequence in context: A263711 A263697 A263746 * A188493 A055292 A035592

Adjacent sequences:  A002521 A002522 A002523 * A002525 A002526 A002527

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Typo in comment corrected by Vaclav Kotesovec, Dec 01 2012

STATUS

approved

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Last modified August 18 14:06 EDT 2017. Contains 290720 sequences.