%I #20 Jul 08 2018 04:05:02
%S 2,3,3,4,3,4,3,5,4,4,3,5,3,4,4,6,3,5,3,5,4,4,3,6,4,4,5,5,3,5,3,7,4,4,
%T 4,6,3,4,4,6,3,5,3,5,5,4,3,7,4,5,4,5,3,6,4,6,4,4,3,6,3,4,5,8,4,5,3,5,
%U 4,5,3,7,3,4,5,5,4,5,3,7,6,4,3,6,4,4,4,6,3,6,4,5,4,4,4,8,3,5,5,6,3,5,3,6
%N Order of rowmotion on the divisor lattice for n.
%C Rowmotion is an action defined on the order ideals of a poset, P, which maps an order ideal I of P to the order ideal generated by the minimal elements of P not contained in I.
%C Conjecture: a(n)=Omega(n)=A001222(n) at n = (p_1^n)(p_2^m)(p_3^2) for primes p_1!=p_2!=p_3! in (Dilks, Pechenik, Striker).
%C Conjecture: a((p_1^n)p_2p_3p_4)=189+27(n-2) for n>=2 and primes p_1!=p_2!=p_3!=p_4!.
%C Diverges from Omega(n)=A001222(n) at n = (p_1^2)(p_2)(p_3)(p_4), (p_1^3)(p_2^3)(p_3^3), (p_1)(p_2)(p_3)(p_4)(p_5) for primes p_1!=p_2!=p_3!=p_4!=p_5, where the values are 189, 33, and 3024, respectively.
%C Conjecture: a(n)!=Omega(n)=A001222(n) when n is not of the form (p_1^r)*(p_2^s)*(p_3^t) with r, s, t >= 0, t<3 or (p_1)(p_2)(p_3)(p_4) for primes p_1!=p_2!=p_3!=p_4.
%H K. Dilks, O. Pechenik, and J. Striker, <a href="https://arxiv.org/abs/1512.00365">Resonance in orbits of plane partitions and increasing tableaux </a>, arXiv preprint arXiv:1512.00365 [math.CO], 2015-2017.
%H K. Dilks, O. Pechenik, and J. Striker, <a href="https://doi.org/10.1016/j.jcta.2016.12.007">Resonance in orbits of plane partitions and increasing tableaux </a>, Journal of Combinatorial Theory, Series A 148 (2017): 244-274.
%H J. Striker, <a href="http://www.ams.org/publications/journals/notices/201706/rnoti-p543.pdf">Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance</a>, Notices of the AMS, June/July 2017, pp. 543-549.
%H J. Striker and N. Williams, <a href="https://arxiv.org/abs/1108.1172">Promotion and Rowmotion</a>, arXiv preprint arXiv:1108.1172 [math.CO], 2011-2012.
%H J. Striker and N. Williams, <a href="https://doi.org/10.1016/j.ejc.2012.05.003">Promotion and Rowmotion</a>, European Journal of Combinatorics 33.8 (2012): 1919-1942.
%F a(n) = Omega(n) = A001222(n) when n is of the form (p_1^r)*(p_2^s)*(p_3^t) with r, s, t >= 0, t<2 as well as (p_1)(p_2)(p_3)(p_4) for primes p_1!=p_2!=p_3!=p_4.
%e a(1)=2 since the only divisor of 1 is 1 which corresponds to a divisor lattice of a single element allowing only for two order ideals, the empty set and {1}, which are contained in a cycle of length 2 under rowmotion.
%e a(3)=3 since the only divisors of 3 are 1 and 3, and thus the corresponding divisor lattice is a chain of 3 elements whose only order ideals are the empty set, {1}, and {1, 3} which are all contained in a single cycle of rowmotion of length 3.
%o (Sage) for n in range (1, 100):; P = posets.DivisorLattice(n); LCM_list(sorted(len(o) for o in P.rowmotion_orbits()))
%Y Cf. A001222.
%K easy,nonn
%O 1,1
%A _Nick Mayers_, _William Franczak_, Jun 08 2018