A166963 Number of permutations in S_{2n} avoiding 123 and 1432 whose matrices are 180-degree symmetric.

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

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: A059078 A335712 A356465 * 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