This is the page for supporting codes for “Birational contractions of $\mathrm{M}}_{0,n}$ 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

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

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'