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A339356 Maximum number of copies of a 123456 permutation pattern in an alternating (or zig-zag) permutation of length n + 9.

%I A339356 #12 Jul 07 2025 04:00:34
%S A339356 16,32,144,256,688,1120,2352,3584,6496,9408,15456,21504,32928,44352,
%T A339356 64416,84480,117744,151008,203632,256256,336336,416416,534352,652288,
%U A339356 821184,990080,1226176,1462272,1785408,2108544,2542656,2976768,3550416,4124064,4870992,5617920,6577648
%N A339356 Maximum number of copies of a 123456 permutation pattern in an alternating (or zig-zag) permutation of length n + 9.
%C A339356 The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.
%H A339356 Lara Pudwell, <a href="https://www.ams.org/journals/notices/202007/rnoti-p994.pdf">From permutation patterns to the periodic table</a>, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.
%H A339356 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
%F A339356 a(2n) = 32*A040977(n-1) = 64*C(n+5,6) - 32*C(n+4,5).
%F A339356 a(2n-1) = 16*A259181(n) = (2*n*(n + 1)*(n + 2)*(n + 3)*(2*n^2 + 6*n + 7))/45.
%F A339356 From _Chai Wah Wu_, Jul 06 2025: (Start)
%F A339356 a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n > 12.
%F A339356 G.f.: x*(-16*x^2 - 16)/((x - 1)^7*(x + 1)^5). (End)
%e A339356 a(1) = 16. The alternating permutation of length 1+9=10 with the maximum number of copies of 123456 is 132547698(10). The sixteen copies are 12468(10), 12469(10), 12478(10), 12479(10), 12568(10), 12569(10), 12578(10), 12579(10), 13468(10), 13469(10), 13478(10), 13479(10), 13568(10), 13569(10), 13578(10), and 13579(10).
%Y A339356 Cf. A168380.
%K A339356 nonn,easy
%O A339356 1,1
%A A339356 _Lara Pudwell_, Dec 01 2020