Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of up-labels to the unique element $y \in L$ which has that set as its down-labels (see [1] and [2]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
References
[1] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[2] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[3] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[4] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[3] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[4] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
def weak_order(n): fcn = lambda p,q : p.permutohedron_lequal(q) return Poset((Permutations(n), fcn)) def lower_labels(P,u): Q = LatticePoset(P) mylist = [] for v in Q.lower_covers(u): my_min = Q.maximal_elements()[0] for j in Q.join_irreducibles(): if Q.join(j,v) == u: if Q.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def upper_labels(P,u): Q = LatticePoset(P) mylist = [] for v in Q.upper_covers(u): my_min = Q.maximal_elements()[0] for j in Q.join_irreducibles(): if Q.join(j,u) == v: if Q.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def mapping(x): n = len(x) P = weak_order(n) u = Permutation(x) ll = lower_labels(P,u) for v in P: if upper_labels(P,v) == ll: return v return None
Properties
bijective
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 15:19 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of up-labels to the unique element $y \in L$ which has that set as its down-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
Diff Description
Any semidistributive lattice L has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element x \in L with a given set of up-labels to the unique element y \in L which has that set as its down-labels (see [12] and [23]). For example, if the lattice is the distributive lattice J(P) of order ideals of a finite poset P, then this reduces to ordinary rowmotion on the order ideals of P.
The weak order (a.k.a. permutohedral order) on the permutations in S_n is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in S_n. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in S_n is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in S_n. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Diff References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations [[DOI:10.1137/140972391]] [[arXiv:1405.6904]]
[2] Barnard, E. The Canonical Join Complex [[DOI:10.37236/7866]] [[arXiv:1610.05137]]
[23] Thomas, H., Williams, N. Rowmotion in slow motion [[DOI:10.1112/plms.12251]] [[arXiv:1712.10123]]
[34] Hopkins, S. The CDE property for skew vexillary permutations [[DOI:10.1016/j.jcta.2019.06.005]] [[arXiv:1811.02404]]
[45] Defant, C., Williams, N. Semidistrim Lattices [[DOI:10.1017/fms.2023.46]] [[arXiv:2111.08122]]
[2] Barnard, E. The Canonical Join Complex [[DOI:10.37236/7866]] [[arXiv:1610.05137]]
[
[
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Sage code
def weak_order(n): fcn = lambda p,q : p.permutohedron_lequal(q) return Poset((Permutations(n), fcn)) def lower_labels(P,ji,u): mylist = [] for v in P.lower_covers(u): my_min = P.maximal_elements()[0] for j in ji: if P.le(j,u) and not(P.le(j,v)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def upper_labels(P,ji,u): mylist = [] for v in P.upper_covers(u): my_min = P.maximal_elements()[0] for j in ji: if P.le(j,v) and not(P.le(j,u)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def mapping(x): n = len(x) P = weak_order(n) ji = LatticePoset(P).join_irreducibles() u = Permutation(x) ll = lower_labels(P,ji,u) for v in P: if upper_labels(P,ji,v) == ll: return v return None
Properties
bijective
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 17:12 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
Diff Description
Any semidistributive lattice L has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element x \in L with a given set of updown-labels to the unique element y \in L which has that set as its downup-labels (see [2] and [3]). For example, if the lattice is the distributive lattice J(P) of order ideals of a finite poset P, then this reduces to ordinary rowmotion on the order ideals of P.
The weak order (a.k.a. permutohedral order) on the permutations in S_n is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in S_n. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in S_n is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in S_n. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
def weak_order(n): fcn = lambda p,q : p.permutohedron_lequal(q) return Poset((Permutations(n), fcn)) def lower_labels(P,ji,u): mylist = [] for v in P.lower_covers(u): my_min = P.maximal_elements()[0] for j in ji: if P.le(j,u) and not(P.le(j,v)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def upper_labels(P,ji,u): mylist = [] for v in P.upper_covers(u): my_min = P.maximal_elements()[0] for j in ji: if P.le(j,v) and not(P.le(j,u)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def mapping(x): n = len(x) P = weak_order(n) ji = LatticePoset(P).join_irreducibles() u = Permutation(x) ll = lower_labels(P,ji,u) for v in P: if upper_labels(P,ji,v) == ll: return v return None
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 17:16 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
def weak_order(n): fcn = lambda p,q : p.permutohedron_lequal(q) return Poset((Permutations(n), fcn)) def lower_labels(P,ji,top,u): mylist = [] for v in P.lower_covers(u): my_min = top for j in ji: if P.le(j,u) and not(P.le(j,v)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def upper_labels(P,ji,top,u): mylist = [] for v in P.upper_covers(u): my_min = top for j in ji: if P.le(j,v) and not(P.le(j,u)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def mapping(x): n = len(x) P = weak_order(n) ji = LatticePoset(P).join_irreducibles() top = P.maximal_elements()[0] u = Permutation(x) ll = lower_labels(P,ji,top,u) for v in P: if upper_labels(P,ji,top,v) == ll: return v return None
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 17:34 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [3].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
@cached_function def weak_order(n): fcn = lambda p,q : p.permutohedron_lequal(q) P = Poset((Permutations(n), fcn)) ji = LatticePoset(P).join_irreducibles() top = P.maximal_elements()[0] return (P,ji,top) def lower_labels(P,ji,top,u): mylist = [] for v in P.lower_covers(u): my_min = top for j in ji: if P.le(j,u) and not(P.le(j,v)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def upper_labels(P,ji,top,u): mylist = [] for v in P.upper_covers(u): my_min = top for j in ji: if P.le(j,v) and not(P.le(j,u)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def mapping(x): n = len(x) (P,ji,top) = weak_order(n) u = Permutation(x) ll = lower_labels(P,ji,top,u) for v in P: if upper_labels(P,ji,top,v) == ll: return v return None
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 17:42 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
Diff Description
Any semidistributive lattice L has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element x \in L with a given set of down-labels to the unique element y \in L which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice J(P) of order ideals of a finite poset P, then this reduces to ordinary rowmotion on the order ideals of P.
The weak order (a.k.a. permutohedral order) on the permutations in S_n is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in S_n. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [34].
The weak order (a.k.a. permutohedral order) on the permutations in S_n is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in S_n. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
@cached_function def weak_order(n): fcn = lambda p,q : p.permutohedron_lequal(q) P = Poset((Permutations(n), fcn)) ji = LatticePoset(P).join_irreducibles() top = P.maximal_elements()[0] return (P,ji,top) def lower_labels(P,ji,top,u): mylist = [] for v in P.lower_covers(u): my_min = top for j in ji: if P.le(j,u) and not(P.le(j,v)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def upper_labels(P,ji,top,u): mylist = [] for v in P.upper_covers(u): my_min = top for j in ji: if P.le(j,v) and not(P.le(j,u)) and P.le(j,my_min): my_min = j mylist.append(my_min) return Set(mylist) def mapping(x): n = len(x) (P,ji,top) = weak_order(n) u = Permutation(x) ll = lower_labels(P,ji,top,u) for v in P: if upper_labels(P,ji,top,v) == ll: return v return None
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 18:09 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
@cached_function def weak_order(n): f = lambda p,q : p.permutohedron_lequal(q) P = Poset((Permutations(n), f)) ji = LatticePoset(P).join_irreducibles() return (P,ji) def lower_labels(P,ji,u): ll = [] for v in P.lower_covers(u): cur_min = None for j in ji: if P.le(j,u) and not(P.le(j,v)) and (cur_min == None or P.le(j,cur_min)): cur_min = j ll.append(cur_min) return Set(ll) def upper_labels(P,ji,u): ul = [] for v in P.upper_covers(u): cur_min = None for j in ji: if P.le(j,v) and not(P.le(j,u)) and (cur_min == None or P.le(j,cur_min)): cur_min = j ul.append(cur_min) return Set(ul) def mapping(x): n = len(x) (P,ji) = weak_order(n) u = Permutation(x) ll = lower_labels(P,ji,u) for v in P: if upper_labels(P,ji,v) == ll: return v return None
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 20:20 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
#note that everything here is dualized (we use dual order and take joins of upper labels) @cached_function def weak_order(n): f = lambda p,q : q.permutohedron_lequal(p) P = LatticePoset(Poset((Permutations(n), f))) ji = P.join_irreducibles() bot = P.minimal_elements()[0] return (P,ji,bot) def lower_labels(P,ji,u): ll = [] for v in P.lower_covers(u): cur_min = None for j in ji: if P.le(j,u) and not(P.le(j,v)) and (cur_min == None or P.le(j,cur_min)): cur_min = j ll.append(cur_min) return ll def upper_labels(P,ji,u): ul = [] for v in P.upper_covers(u): cur_min = None for j in ji: if P.le(j,v) and not(P.le(j,u)) and (cur_min == None or P.le(j,cur_min)): cur_min = j ul.append(cur_min) return ul def mapping(x): n = len(x) (P,ji,bot) = weak_order(n) u = Permutation(x) ll = upper_labels(P,ji,u) v = bot for w in ll: v = P.join(v,w) return v
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 20:39 by Sam Hopkins
Identifier
Mp00000:
Permutations
—Weak order rowmotion⟶
Permutations
Name
Weak order rowmotion
Description
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$. Not much is known about the dynamics of weak order rowmotion, which seems unpredictable. One nontrivial collection of homomesies is described in Corollary 6.14 of [4].
References
[1] Reading, N. Noncrossing Arc Diagrams and Canonical Join Representations DOI:10.1137/140972391 arXiv:1405.6904
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
[2] Barnard, E. The Canonical Join Complex DOI:10.37236/7866 arXiv:1610.05137
[3] Thomas, H., Williams, N. Rowmotion in slow motion DOI:10.1112/plms.12251 arXiv:1712.10123
[4] Hopkins, S. The CDE property for skew vexillary permutations DOI:10.1016/j.jcta.2019.06.005 arXiv:1811.02404
[5] Defant, C., Williams, N. Semidistrim Lattices DOI:10.1017/fms.2023.46 arXiv:2111.08122
Sage code
#note that everything here is dualized (we use dual order and take joins of upper labels) @cached_function def dual_weak_order(n): f = lambda p,q : q.permutohedron_lequal(p) P = LatticePoset(Poset((Permutations(n), f))) ji = P.join_irreducibles() bot = P.minimal_elements()[0] return (P,ji,bot) def upper_labels(P,ji,u): ul = [] for v in P.upper_covers(u): cur_min = None for j in ji: if P.le(j,v) and not(P.le(j,u)) and (cur_min == None or P.le(j,cur_min)): cur_min = j ul.append(cur_min) return ul def mapping(x): (P,ji,bot) = dual_weak_order(len(x)) v = bot for w in upper_labels(P,ji,Permutation(x)): v = P.join(v,w) return v
Properties
bijective, graded
Created
Mar 02, 2024 at 15:19 by Sam Hopkins
Updated
Mar 02, 2024 at 20:49 by Sam Hopkins
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