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%I #8 Jan 11 2024 09:07:18
%S 32,64,320,576,1696,2816,6400,9984,19392,28800,50304,71808,116160,
%T 160512,244992,329472,480480,631488,887744,1144000,1560416,1976832,
%U 2629120,3281408,4271488,5261568,6723840,8186112,10294656,12403200,15379968,18356736,22480800,26604864
%N Maximum number of copies of a 1234567 permutation pattern in an alternating (or zig-zag) permutation of length n + 11.
%C The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.
%H 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.
%F a(2n) = 64*A050486(n-1) = 128*C(n+6,7) - 64*C(n+5,6).
%F a(2n-1) = 128*C(n+4,7) + 128*C(n+4,6) + 32*C(n+4,5).
%F D-finite with recurrence (-n+1)*a(n) +2*a(n-1) +16*a(n-2) +2*a(n-3) +(n+7)*a(n-4)=0. - _R. J. Mathar_, Jan 11 2024
%e a(1) = 32. The alternating permutation of length 1+11=12 with the maximum number of copies of 1234567 is 132547698(11)(10)(12). The 32 copies are 12468(10)(12), 12469(10)(12), 12478(10)(12), 12479(10)(12), 12568(10)(12), 12569(10)(12), 12578(10)(12), 12579(10)(12), 13468(10)(12), 13469(10)(12), 13478(10)(12), 13479(10)(12), 13568(10)(12), 13569(10)(12), 13578(10)(12), 13579(10)(12), 12468(11)(12), 12469(11)(12), 12478(11)(12), 12479(11)(12), 12568(11)(12), 12569(11)(12), 12578(11)(12), 12579(11)(12), 13468(11)(12), 13469(11)(12), 13478(11)(12), 13479(11)(12), 13568(11)(12), 13569(11)(12), 13578(11)(12), and 13579(11)(12).
%p A339358 := proc(n)
%p nhalf := ceil(n/2) ;
%p if type(n,'even') then
%p 128*binomial(nhalf+6,7)-64*binomial(nhalf+5,6) ;
%p else
%p 128*binomial(nhalf+4,7)+128*binomial(nhalf+4,6)+32*binomial(nhalf+4,5) ;
%p end if;
%p end proc:
%p seq(A339358(n),n=1..40) ; # _R. J. Mathar_, Jan 11 2024
%Y Cf. A168380.
%K nonn
%O 1,1
%A _Lara Pudwell_, Dec 01 2020
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