Blancmange Curve - Relation To Simplicial Complexes

Relation To Simplicial Complexes

Let

$N=\binom{n_t}{t}+\binom{n_{t-1}}{t-1}+\ldots+\binom{n_j}{j},\quad n_t > n_{t-1} > \ldots > n_j \geq j\geq 1.$

Define the Kruskal–Katona function

$\kappa_t(N)={n_t \choose t+1} + {n_{t-1} \choose t} + \dots + {n_j \choose j+1}.$

The Kruskal–Katona theorem states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes.

As t and N approach infinity, (suitably normalized) approaches the blancmange curve.

Read more about this topic: Blancmange Curve

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