This is a page for supporting codes for “Birational contractions of \(\overline{\mathrm{M}}_{0,n}\) and combinatorics of extremal assignments”, written by the third author.
Ordinary extremal assignments (Section 6)
A Sage code finding the smallest extremal assignment with prescribed assigned vertices (You may also download the code here.)
The command findSmallestFamily(n, list) finds the smallest collection of set partitions satisfying the assumption in Theorem 1.2.
- n: number of labels
- list: list of set partitions. A set partition is a list of lists.
For instance, an extremal assignment generated by central vertices of stars corresponding to [[1, 2, 3], [4, 5, 6], [7, 8]] and [[1, 2], [3], [4, 5, 6, 7, 8]] is an atomic extremal assignment generated by a star associated to [[1, 2], [3], [4, 5, 6], [7, 8]].
findSmallestFamily(8, [[[1,2,3],[4,5,6],[7,8]],[[1,2],[3],[4,5,6,7,8]]])
[[[1, 2], [3], [4, 5, 6], [7, 8]]]
If given collection of set partitions already satisfies the condition in (3) of Theorem 1.1, it returns the initial collection.
findSmallestFamily(8,[[[1,2,3],[4,5,6],[7,8]],[[1,2,3,4],[5,6],[7,8]]])
[[[1, 2, 3], [4, 5, 6], [7, 8]], [[1, 2, 3, 4], [5, 6], [7, 8]]]
If the tight upper bound of two corruptions is the complete partition, then it returns “Not possible”.
findSmallestFamily(8,[[[1],[2],[3],[4],[5],[6,7,8]],[[1,2,3,4],[5],[6],[7],[8]]])
Not possible
\(S_n\)-invariant extremal assignments (Section 9)
A Sage code finding the smallest \(S_n\)-invariant extremal assignment with prescribed assigned vertices (You may also download the code here.)
The command findSmallestFamily(n, list) calculates the smallest special family of integer partitions which must be assigned in the smallest \(S_n\)-invariant extremal assignment containing given integer partitions.
- n: number of labels
- list: list of integer partitions. An inter partition is also a list of positive integers whose sum is n.
findSmallestFamily(14, [[5,4,3,2]])
[[3, 3, 3, 3, 2], [3, 3, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2]]
Thus if \(Z\) is an \(S_n\)-invariant extremal assignment assigning the central vertex of a star associated to the partition [[5, 4, 3, 2]], then it must assign central vertices of three stars associated to [3, 3, 3, 3, 2], [3, 3, 2, 2, 2, 2], and [2, 2, 2, 2, 2, 2, 2].
The following example shows that a union of special partitions may not a special family.
findSmallestFamily(13, [[3,3,3,2,2]])
[[3, 3, 3, 2, 2]]
findSmallestFamily(13, [[7,2,2,2]])
[[7, 2, 2, 2]]
findSmallestFamily(13, [[3,3,3,2,2],[7,2,2,2]])
[[3, 3, 3, 2, 2], [3, 2, 2, 2, 2, 2]]
If the smallest special family has the complete partition [[1, 1, 1, … , 1]], so there is no \(S_n\)-invariant extremal assignment, it returns ‘Not possible’.
findSmallestFamily(10, [[4,3,2,1]])
'Not possible'