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A339358
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Maximum number of copies of a 1234567 permutation pattern in an alternating (or zig-zag) permutation of length n + 11.
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1
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32, 64, 320, 576, 1696, 2816, 6400, 9984, 19392, 28800, 50304, 71808, 116160, 160512, 244992, 329472, 480480, 631488, 887744, 1144000, 1560416, 1976832, 2629120, 3281408, 4271488, 5261568, 6723840, 8186112, 10294656, 12403200, 15379968, 18356736, 22480800, 26604864
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OFFSET
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1,1
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COMMENTS
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The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.
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LINKS
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FORMULA
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a(2n) = 64*A050486(n-1) = 128*C(n+6,7) - 64*C(n+5,6).
a(2n-1) = 128*C(n+4,7) + 128*C(n+4,6) + 32*C(n+4,5).
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
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EXAMPLE
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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).
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MAPLE
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nhalf := ceil(n/2) ;
if type(n, 'even') then
128*binomial(nhalf+6, 7)-64*binomial(nhalf+5, 6) ;
else
128*binomial(nhalf+4, 7)+128*binomial(nhalf+4, 6)+32*binomial(nhalf+4, 5) ;
end if;
end proc:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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