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A339358
Maximum number of copies of a 1234567 permutation pattern in an alternating (or zig-zag) permutation of length n + 11.
1
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
OFFSET
1,1
COMMENTS
The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.
LINKS
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.
FORMULA
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
EXAMPLE
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).
MAPLE
A339358 := proc(n)
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:
seq(A339358(n), n=1..40) ; # R. J. Mathar, Jan 11 2024
CROSSREFS
Cf. A168380.
Sequence in context: A256819 A358250 A235057 * A249116 A110562 A275187
KEYWORD
nonn
AUTHOR
Lara Pudwell, Dec 01 2020
STATUS
approved