A338429 Maximum number of copies of a 1234 permutation pattern in an alternating (or zig-zag) permutation of length n + 5.

4, 8, 28, 48, 104, 160, 280, 400, 620, 840, 1204, 1568, 2128, 2688, 3504, 4320, 5460, 6600, 8140, 9680, 11704, 13728, 16328, 18928, 22204, 25480, 29540, 33600, 38560, 43520, 49504, 55488, 62628, 69768, 78204, 86640, 96520, 106400, 117880, 129360

Offset: 1

Author

Lara Pudwell, Dec 01 2020

Keywords

• nonn
• easy
• changed

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) = 4. The alternating permutation of length 1+5=6 with the maximum number of copies of 1234 is 132546.  The four copies are 1246, 1256, 1346, and 1356.
a(2) = 8. The alternating permutation of length 2+5=7 with the maximum number of copies of 1234 is 1325476. The eight copies are 1246, 1256, 1247, 1257, 1346, 1356, 1347, and 1357.
		

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 (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A006325, A072819, A168380.

Formula

a(2n) = A072819(n+1) = (2*n*(n + 2)*(n + 1)^2)/3.

a(2n-1) = 4*A006325(n+1) = (2*n*(n + 1)*(n^2 + n + 1))/3.

G.f.: 4*x*(1 + x^2)/((1 - x)^5*(1 + x)^3). - Stefano Spezia, Dec 12 2021

E.g.f.: (x*(75 + 51*x + 14*x^2 + x^3)*cosh(x) + (21 + 45*x + 57*x^2 + 14*x^3 + x^4)*sinh(x))/24. - Stefano Spezia, Aug 29 2025