Reference no: EM131085710
Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-
(a) Let L be a finite lattice and let f(x, s) be a C-valued function defined for all x, s ∈ L. Set F(x, s) = ∑z≤x f(z, s). Show that
det[F(x ∧ y, x)]x,y∈L = ∏x∈L f(x, x),
and use this to show that
det[gcd(i, j)]i,j=1n = k=1∏nφ(k).
Hint: Define the matrix M = M(x, y) whose entries are ζ(x, y)f(x, y). Investigate M and MT.
(b) Let G be a graph. For any positive integer k, let χ(k) be the number of proper k-colorings of G (i.e. the number of functions c: V (G) → [k] such that c(u) ≠ c(v) for any uv ∈ E(G)). Let LG be the poset of all partitions π of V(G) such that the induced subgraph on every block of π is connected (with ordering by refinement).
Show that
χ(n) = ∑π∈L_Gµ(0ˆ, π)n|π|.
where |π| is the number of blocks of π, and µ is the Mobius function of LG.
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