%I #31 Aug 29 2025 08:06:01

%S 4,8,28,48,104,160,280,400,620,840,1204,1568,2128,2688,3504,4320,5460,

%T 6600,8140,9680,11704,13728,16328,18928,22204,25480,29540,33600,38560,

%U 43520,49504,55488,62628,69768,78204,86640,96520,106400,117880,129360

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

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

%H Lara Pudwell, <a href="https://www.ams.org/journals/notices/202007/rnoti-p994.pdf">From permutation patterns to the periodic table</a>, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).

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

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

%F G.f.: 4*x*(1 + x^2)/((1 - x)^5*(1 + x)^3). - _Stefano Spezia_, Dec 12 2021

%F 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

%e 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.

%e 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.

%Y Cf. A006325, A072819, A168380.

%K nonn,easy

%O 1,1

%A _Lara Pudwell_, Dec 01 2020