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# shellingOrder -- finds a shelling of a simplicial complex, if one exists

## Synopsis

• Usage:
L = shellingOrder S
• Inputs:
• Optional inputs:
• Permutation => a list, default value {}, of integers from $0$ to one less than the number of facets which is applied to the facets before the recursive search for a shelling is executed.
• Random => , default value false, whether to use a random permutation to the facet list before the recursive search for a shelling is executed.
• Outputs:
• L, a list, a shelling order of the facets of $S$, if one exists

## Description

If $S$ is pure, then definition III.2.1 in [St] is used. That is, $S$ is shellable if its facets can be ordered $F_1, .., F_n$ so that the difference in the $j$-th and $j-1$-th subcomplex has a unique minimal face, for $2 \leq j \leq n$.

If $S$ is non-pure, then definition 2.1 in [BW-1] is used. That is, a simplicial complex $S$ is shellable if the facets of $S$ can be ordered $F_1, .., F_n$ such that the intersection of the faces of the first $j-1$ with the faces of the $F_j$ is pure and $dim F_j - 1$-dimensional.

This function attempts to build up a shelling order of $S$ recursively. In particular, a depth-first search is used to attempt to build up a shelling order from the bottom, that is, from the first facet in the order.

In the case when $S$ is non-pure, then the search is restricted to the maximal dimension facets remaining to be added. This allows a shelling order in reverse dimension order to be returned whenever $S$ is indeed shellable.

 i1 : R = QQ[a..f]; i2 : shellingOrder simplicialComplex {a*b*c*d*e} o2 = {a*b*c*d*e} o2 : List i3 : shellingOrder simplicialComplex {a*b*c, b*c*d, c*d*e} o3 = {c*d*e, b*c*d, a*b*c} o3 : List i4 : shellingOrder simplicialComplex {a*b*c, c*d*e} o4 = {} o4 : List i5 : shellingOrder simplicialComplex {a*b*c, c*d, d*e, e*f, d*f} o5 = {a*b*c, c*d, d*e, d*f, e*f} o5 : List i6 : shellingOrder simplicialComplex {a*b*c, c*d, d*e*f} o6 = {} o6 : List

The options Random and Permutation can be used to try to find alternate shelling orders. Random applies a random permutation to the facet list and Permutation applies a supplied permutation to the list. In the non-pure case, the facets are subsequently ordered in reverse dimension order but retaining the ordering within dimensions.

The options Random and Permutation are mutually exclusive.

 i7 : S = simplicialComplex {a*b*c, b*c*d, c*d*e, d*e*f}; i8 : shellingOrder S o8 = {d*e*f, c*d*e, b*c*d, a*b*c} o8 : List i9 : shellingOrder(S, Random => true) o9 = {b*c*d, a*b*c, c*d*e, d*e*f} o9 : List i10 : shellingOrder(S, Permutation => {3,2,1,0}) o10 = {a*b*c, b*c*d, c*d*e, d*e*f} o10 : List

The shelling order is cached if it exists, however, if either option is used, then the cache is ignored.