Twenty-six sets of permutations from Schubert varieties and Schubert polynomials. Twenty-three are permutation classes, nineteen of them with a combinatorial specification (seven of these not on PermPAL yet); three are sets that no classical patterns describe. Type a permutation to see which of them it belongs to, where it sits among them, and how each specification builds it.
The hierarchy of classes
Each line goes up from a set to a larger one. Between two classes the inclusion comes from their bases, and with a set from a theorem. Hover over a line to see where it comes from. Schubert varieties Xw are indexed as Lakshmibai and Sandhya do, so that the smooth ones avoid 3412 and 4231.
contains wdoes not contain wnot a permutation classno specification yetabout Schubert polynomialsClick a set for its card below.
Test a permutation
One-line notation, like 615342. For length 10 or more, put a dot between the entries, like 4.5.6.7.8.9.10.1.2.3 (spaces work too).
Try
Showpieces
How w lives in the specification
Compare two sets
Is one inside the other? If it is, why; if it is not, the smallest permutation that shows it, in the one and not in the other: the first of the shortest length, in lexicographic order, among all permutations up to length 9. Every two of the sets have one or the other.
All pairs at a glance
Row A, column B: ⊆ when A is inside B, and otherwise the length of the smallest permutation in A and not in B. Click a cell to compare the two.
The classes and sets
A class PermPAL has gets a short card with a link. The others get a full card: the classes with a specification that PermPAL does not have yet, the classes with none yet, and the sets that are not classes.