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Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Thomas Britz, Peter J. Cameron
A third way to generalize Greene's theorem and the critical theorem is to replace matroids by more general structures. Helgason and Whittle proved a polymatroid generalization of the critical theorem (see [731, 1160, 1162]); Ardila [32] provided hyperplane arrangement generalizations; Farr [474–476] and Britz and Shiromoto [218] proved purely set-theoretical generalizations of Greene's theorem; and similar earlier work is described in the excellent exposition by Kung [731]. As described in Section 16.5, Cameron [262] found a polynomial that simultaneously generalizes the Tutte polynomial and the cycle index of a certain group; see also [263, 264, 266, 267, 975].
Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
1|G|[mcyc(π1)+mcyc(π2)+…] $$ \frac{1}{{|G|}}[m^{{cyc(\pi 1)}} + m^{{cyc(\pi 2)}} + \ldots ] $$ Polya’s theorem: Let G be a group of permutations on a set A with cycle index PG(x1,x2,…,xk) $ P_{G} (x_{1} , x_{2} , \ldots , x_{k} ) $ . Let C(A,R) $ C(A, R) $ be the collection of all colorings of A using colors in R. If w is a weight assignment on R, then the pattern inventory of colorings in C(A,R) $ C(A, R) $ is given by
In Many Circles. Permutations as Products of Cycles.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
Summing all these monomials over all n-permutations, we get the augmented cycle indexZ˜(Sn) of the symmetric group Sn.
Method of correcting stitching errors in reconstructing a synthetic-aperture digital hologram with seams
Published in Journal of Modern Optics, 2019
Meng Ding, Qi Fan, Yin Su, Baiyu Yang, Changhui Tian, Binke Wang, Yunfei Wang
The correction holograms are reconstructed by the method described in 2.1. In the simulation experiment, the cycle index is 200, ϵ is 1.76940 × 10−4, the initial phase value is set with the phase of the converging spherical wave on CCD array 1, and the weight value α is 1.2.