Since Good’s paper [5j appeared in 1960, :i lot of other works have been published on this theme. Good’s proof of his theorem being analytical, <:he later authors considered Lagrange inversion within the theory of formal power series. The first attempt for a purely combinatorial proof was made by Chottin [ 1 j, who treated a special case of the two dimensional formula. Tutte gave an extensive d...

We construct a spectral sequence converging to the cohomology with compact support of m-th contact locus complex polynomial. The first page is explicitly described in terms log resolution and coincides McLean’s Floer $m$‑th iterate monodromy, when polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs holomorphic functions are embedded topologically...

we find an explicit expression of the tutte polynomial of an $n$-fan. we also find a formula of the tutte polynomial of an $n$-wheel in terms of the tutte polynomial of $n$-fans. finally, we give an alternative expression of the tutte polynomial of an $n$-wheel and then prove the explicit formula for the tutte polynomial of an $n$-wheel.

A polynomial is unimodal if its sequence of coefficients are increasing up to an index, and then are decreasing after that index. A polynomial is logconcave if the sequence of the logarithms of the coefficients is concave. We prove that if P (x) is a polynomial with nonnegative non-decreasing coefficients then P (x+z) is unimodal for any natural z. Furthermore, we prove that if P (x) is a log-c...

We have found that there are more than a dozen classical generating functions that could be suitably symbolized to yield various symbolic sum formulas by employing Mullin-Rota’s theory of binomial enumeration. Various special formulas and identities involving well-known number sequences or polynomial sequences are presented as illustrative examples. The convergence of the symbolic summations is...

In this article we study weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in L. We find a necessary condition and show that for any bounded measurable function φ on an ergodic system, the sequence φ(Tnx) is universally good for almost every x. The linear case was cov...

We generalize the colored Jones polynomial to 4-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman–Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.

Splines are piecewise polynomial functions that have certain “regularity” properties. These can be defined on all finite intervals, and intervals of the form (−∞, a], [b,∞) or (−∞,∞). We have already encountered linear splines, which are simply continuous, piecewise-linear functions. More general splines are defined similarly to the linear ones. They are labeled by three things: (1) a knot sequ...

Polynomial-clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone C, a sequence B ∈ X is C-reducible to A ∈ X if there is an algorithm that computes B from A using only effectively selected functio...