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.
Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
FORMULA
a(2n) = 32*A040977(n-1) = 64*C(n+5,6) - 32*C(n+4,5).
a(2n-1) = 16*A259181(n) = (2*n*(n + 1)*(n + 2)*(n + 3)*(2*n^2 + 6*n + 7))/45.
From Chai Wah Wu, Jul 06 2025: (Start)
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.
G.f.: x*(-16*x^2 - 16)/((x - 1)^7*(x + 1)^5). (End)
EXAMPLE
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).
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lara Pudwell, Dec 01 2020
STATUS
approved
