A319615 Maximum orbit size of rowmotion on the symmetric group.

1, 2, 4, 12, 20, 128, 412, 2184

Offset

1,2

Comments

Rowmotion on a semidistributive lattice is obtained by sending an element with a specified join-canonical decomposition to the unique element with the same meet-canonical decomposition (under the bijection between join- and meet-irreducible elements). a(n) is the maximum orbit size of rowmotion on the symmetric group.

Links

Table of n, a(n) for n=1..8.

Emily Barnard, The canonical join complex, arXiv preprint arXiv:1610.05137 [math.CO], 2016.

Hugh Thomas and Nathan Williams, Rowmotion in slow motion, arXiv preprint arXiv:1712.10123 [math.CO], 2017-2018. See 7.11 p. 26.

Hugh Thomas and Nathan Williams, Independence Posets, arXiv preprint arXiv:1805.00815 [math.CO], 2018.

Example

a(3)=4 because of the orbit [213, 312, 132, 231].
a(4)=12 because of the orbit [1342, 3241, 2314, 4213, 1243, 3421, 3124, 4132, 1423, 2431, 2134, 4312].

Prog

(Sage)
from sage.combinat.cyclic_sieving_phenomenon import orbit_decomposition
def AA(n):
    L=WeylGroup(['A', n]).weak_lattice()
    lower=dict([(l, Set(L.canonical_joinands(l))) for l in L])
    J, M=L.join_irreducibles(), L.meet_irreducibles()
    meet_to_join=dict([(m, L.meet([l for l in J if L.le(l, L.upper_covers(m)[0]) and not(L.le(l, m))])) for m in M])
    upper=dict([(Set([meet_to_join[i] for i in L.canonical_meetands(l)]), l) for l in L])
    return max(map(len, orbit_decomposition(L, lambda w: upper[lower[w]] )))

Crossrefs

Sequence in context: A121569 A319213 A099603 * A343865 A062767 A173650

Adjacent sequences: A319612 A319613 A319614 * A319616 A319617 A319618

Keyword

nonn,hard,more

Author

Nathan Williams, Sep 24 2018

Extensions

a(7) from Lucas A. Brown, May 29 2023

Status

approved