A305716 Order of rowmotion on the divisor lattice for n.

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, 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, 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

Offset

1,1

Comments

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.

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).

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!.

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.

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.

Links

Table of n, a(n) for n=1..104.

K. Dilks, O. Pechenik, and J. Striker, Resonance in orbits of plane partitions and increasing tableaux , arXiv preprint arXiv:1512.00365 [math.CO], 2015-2017.

K. Dilks, O. Pechenik, and J. Striker, Resonance in orbits of plane partitions and increasing tableaux , Journal of Combinatorial Theory, Series A 148 (2017): 244-274.

J. Striker, Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance, Notices of the AMS, June/July 2017, pp. 543-549.

J. Striker and N. Williams, Promotion and Rowmotion, arXiv preprint arXiv:1108.1172 [math.CO], 2011-2012.

J. Striker and N. Williams, Promotion and Rowmotion, European Journal of Combinatorics 33.8 (2012): 1919-1942.

Formula

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.

Example

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.
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.

Prog

(Sage) for n in range (1, 100):; P = posets.DivisorLattice(n); LCM_list(sorted(len(o) for o in P.rowmotion_orbits()))

Crossrefs

Cf. A001222.

Sequence in context: A030349 A285203 A085887 * A297616 A213251 A049108

Adjacent sequences: A305713 A305714 A305715 * A305717 A305718 A305719

Keyword

easy,nonn

Author

Nick Mayers, William Franczak, Jun 08 2018

Status

approved