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A166963 Number of permutations in S_{2n} avoiding 123 and 1432 whose matrices are 180-degree symmetric. 1
1, 2, 6, 12, 27, 61, 138, 309, 694, 1560, 3506, 7877, 17699, 39770, 89363, 200796, 451184, 1013802, 2277993, 5118603, 11501396, 25843403, 58069600, 130481206, 293188608, 658788823, 1480285049 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Trivially, this also counts 180-degree symmetric permutations avoiding 321 and 4123, 123 and 3214, or 321 and 2341. For the other 140 pairs of patterns in S_3 and S_4, the sequence of symmetric permutations avoiding those patterns is either finite (as in 123 and 4321, by Erdos-Szekeres) or counted by an easily-recognized sequence such as alternating Fibonacci numbers, Catalan numbers, squares plus one, or the naturals.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

FORMULA

a(n) = 2a(n-1) + a(n-3) + a(n-4) - a(n-5).

G.f.: (-x^3 + 2x^2 + 1)/(x^5 - x^4 - x^3 - 2x + 1).

EXAMPLE

For n=2, the a(2) = 6 solutions are 2143, 2413, 3142, 3412, 4231, and 4321. The two other 180-degree symmetric permutations in S_4 are 1234 and 1324, both of which contain the pattern 123.

MATHEMATICA

LinearRecurrence[{2, 0, 1, 1, -1}, {1, 2, 6, 12, 27}, 50] (* G. C. Greubel, May 29 2016 *)

CROSSREFS

Sequence in context: A091919 A059078 A335712 * A188476 A155583 A140853

Adjacent sequences:  A166960 A166961 A166962 * A166964 A166965 A166966

KEYWORD

nonn

AUTHOR

David Lonoff and Jonah Ostroff (jonah.ostroff(AT)gmail.com), Oct 25 2009

EXTENSIONS

Fixed typos caused by non-ASCII symbol Jonah Ostroff (jonah.ostroff(AT)gmail.com), Oct 25 2009

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)