A339358 Maximum number of copies of a 1234567 permutation pattern in an alternating (or zig-zag) permutation of length n + 11.

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

Author

Lara Pudwell, Dec 01 2020

Keywords

• nonn

Comments

The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.

Examples

			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).
		

Links

• Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.

Crossrefs

Cf. A168380.

Programs

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

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