Abstract simplicial complex: Difference between revisions

Revision as of 19:45, 28 November 2022

Revision as of 09:23, 10 July 2023

Line 82:

==Enumeration==

The number of abstract simplicial complexes on up to ''n'' labeled elements (that is on a set ''S'' of size ''n'') is one less than the ''n''th [[Dedekind number]]. These numbers grow very rapidly, and are known only for {{math|''n'' ≤ 89}}; the Dedekind numbers are (starting with ''n'' = 0):

:1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365 {{OEIS|id=A014466}}. This corresponds to the number of non-empty [[antichain]]s of subsets of an {{math| ''n''}} set.

The number of abstract simplicial complexes whose vertices are exactly ''n'' labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993" {{OEIS|id=A006126}}, starting at ''n'' = 1. This corresponds to the number of antichain covers of a labeled ''n''-set; there is a clear bijection between antichain covers of an ''n''-set and simplicial complexes on ''n'' elements described in terms of their maximal faces.

The number of abstract simplicial complexes on exactly ''n'' unlabeled elements is given by the sequence "1, 2, 5, 20, 180, 16143" {{OEIS|id=A006602}}, starting at ''n'' = 1.