Eulerian Numbers by T. Kyle Petersen

Author:T. Kyle Petersen
Language: eng
Format: epub, pdf
Publisher: Springer New York, New York, NY

Using the same idea for all the faces, the fine Hilbert series for this example is

Passing to the usual Hilbert series, we have:

In Section 8.​8 we found the f-vector of Δ is (1, 5, 6, 1) and the h-vector is , and now we see these coefficients appearing again. This is not a coincidence.

In general for a face F of a simplicial complex Δ, we get:

Hence, we can write:

and so if we set x i  = t to get the ordinary Hilbert series, we have

where the second equality follows because and the third equality follows from the transformation from f- to h-polynomials given in Equation 8.2. The main point for us is the observation that the numerator of the Hilbert series of the Stanley-Reisner ring is the h-polynomial.

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