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

1,1

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

Table of n, a(n) for n=1..40.

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).

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

Example

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.

Crossrefs

Cf. A006325, A072819, A168380.

Sequence in context: A110132 A099513 A104042 * A117864 A020138 A306448

Adjacent sequences: A338426 A338427 A338428 * A338430 A338431 A338432

Keyword

nonn,easy

Author

Lara Pudwell, Dec 01 2020

Status

approved