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# Tag Archives: Catalan numbers

## An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

(This is a duplicate of a Mathoverflow question posed in Oct 2014 that ran the gauntlet of the OCD cadre there–the demonstrative ones I assume avoid stepping on cracks in the sidewalk and become obstructive, hostile, and/or jealous at popular … Continue reading

## Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion

The raising op for any Appell sequence is determined by the derivative of the log of the e.g.f. of the basic number sequence, connecting the op to the combinatorics of the cumulant expansion OEIS-127671 of the moment generating function and … Continue reading

Posted in Math
Tagged Appell sequences, Associahedra, Bernoulli, Bernoulli polynomials, Catalan numbers, Compositional inverse, Cumulants, Dyck lattce paths, Eulerian numbers, Free cumulants, Free probability, Hirzebruch Todd class criterion, Lagrange inversion, Noncrossing partitions, Permutohedra, Raising operators, Riemann zeta
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## Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra

Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra: A brief note (pdf) on some relations among these topics, including the Catalan and Fuss-Catalan numbers. References are provided linking the analysis with the distribution of eigenvalues of random … Continue reading

## Depressed Equations and Generalized Catalan Numbers

Discriminating Deltas, Depressed Equations, and Generalized Catalan NumbersĀ is a set of notes on the the relation of generating functions of the generalized Catalan numbers, e.g., OEIS-A001764, to the compositional inverse of and the tangent envelope of associated discriminant curves. Added … Continue reading

Posted in Math
Tagged Catalan numbers, Compositional inversion, Determinants, Dirac delta function, Discriminant curves and tangent envelope, Discriminants, Functional equations for Fuss-Catalan numbers, Fuss-Catalan numbers, Generalized Catalan numbers, Graphical solutions of depressed equations, Lagrange inversion, Laplace transform, Legendre transform, Power series
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