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%I #56 May 04 2023 10:04:45
%S 1,1,1,1,2,3,6,11,22,44,89,185,382,808,1702,3635,7779,16736,36229,
%T 78466,171238,373203,819186,1795611,3958662,8721086,19294525,42691298,
%U 94733886,210379132,468084856,1042703207,2325575076,5193931583,11609749877,25986720374,58203955771
%N Number of parity-alternating permutations of [n] avoiding the pattern 321.
%C A permutation is parity-alternating if it sends odd integers to odd integers, and even integers to even integers. It avoids 321 if there is no subsequence a..b..c with a > b > c. The values are computed by Michael Albert, see MathOverflow link.
%H Peter J. Taylor, <a href="/A354208/b354208.txt">Table of n, a(n) for n = 0..50</a>
%H Per Alexandersson, Samuel Asefa Fufa, Frether Getachew and Dun Qiu, <a href="https://arxiv.org/abs/2201.08168">Pattern-avoidance and Fuss-Catalan numbers</a>, arXiv:2201.08168 [math.CO], 2022. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Getachew/get3.html">J. Int. Seq.</a> (2023) Vol. 26, Art. 23.4.2.
%H MathOverflow, <a href="https://mathoverflow.net/questions/424040/321-avoiding-and-parity-alternating-permutations">321-avoiding and parity-alternating permutations</a>, Jun 06 2022.
%H Nathan Williams, <a href="https://personal.utdallas.edu/~nxw170830/docs/Papers/Misc/Oberwolfach_EC_2022_problem_session.pdf">Oberwolfach Problem Session: Enumerative Combinatorics 2022</a>, Univ. Texas Dallas (2023). See example 3.4, where this sequence is misidentified by typographical error.
%e For n=4, the two permutations are 1234, 3412.
%e For n=5, we have 12345, 34125, 14523.
%e For n=6, we have 123456, 341256, 145236, 125634, 561234, 345612.
%Y Cf. A000108 (321-avoiding permutations), A010551 (parity-alternating permutations).
%K nonn
%O 0,5
%A _Per W. Alexandersson_, Jun 06 2022
%E Offset corrected and terms a(30) and beyond from _Peter J. Taylor_, Jun 10 2022
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