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8h
asked Semigroup nilpotents and compostional inversion
1d
comment A number array related to colored necklaces and the primes
Yep. Thanks. For me, it's important to note that the totient gives the degree of the cyclotomic polynomials and that's why I was able to notice the regularity for the prime rows in analogy with that of the cyclotomic polynoms.
1d
comment A number array related to colored necklaces and the primes
Very neat. These little arithmetics are like hummingbirds to me--they don't visit my garden often, but when they do, I'm enchanted.
1d
accepted A number array related to colored necklaces and the primes
2d
asked A number array related to colored necklaces and the primes
2d
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
See the cyclotomic identity on Wikipedia for a reciprocal relation between the GS for Assoc and free-Lie algebras. Reciprocals can be related to permutohedra which can be morphed into associahedra.
2d
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
The GS for Assoc is $x/(1-x)$. Umbralized with $f(x)=c.x/(1-c.x)=c_1x + c_2x^2+...$, $exp(t f(x))$ is the gen. for the Lah partition polynoms. Integrating the exp and expressing as a Laplace trf. gives the Lagrange partition polynoms for inversion, which can be expressed as colored binary trees related to $x^2d/dx$. F. Chapoton (F.C.) has written extensively on operads and the alg. comb of trees, and associahedra. He could no doubt spell out the connections as well as flesh out the items in my list and extend it. Maybe he'll weigh in.
Oct
20
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Composites are noted in A & S (see partitions), but generally you are right. Around 1857, Graves wrote down the relation between iterated derivatives and inversion (no expansion?). Maybe it wasn't until Boltzmann intro. stat mech and Feynman, his diagrams that combinatorics really started to be noticed in physics (aside from the dalliance with knots and vortices/atoms) whereas special fcts. had been around in a central role. When and why Bruno-like partitions nudged their way onto the stage, I don'know (maybe cumulants and cluster expansions in stat mech), but they are in A&S.
Oct
20
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Given Cayley's familiarity with Catalans as a generating series for enumerating certain forests of trees and the dissections of polygons, his use of analytic trees to model iterated derivatives (operandators), and his, Sylvester's, and Graves' work on normal ordering of differential operators, it's surprising that he didn't discover these numbers through the connection between iterated derivatives and Lagrange inversion, which he frequently used. (Or maybe he did.)
Oct
20
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Added Hopf
Oct
20
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Added links
Oct
20
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
added 161 characters in body
Oct
20
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
added 161 characters in body
Oct
20
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
The simplicial dual and the h-vectors also pop up in interesting places, and if you re-scale the indeterminates of the o.g.f. for the Lagrange inversion to generate an e.g.f. formulation, the Whitehouse simplicial complexes, tropical Grassmannians, and phylogenetic trees sprout up (expressing the inversion in terms of the indeterminates of the reciprocal of the function introduces the Narayana h-vectors), so it's a skip and a hopf from the associahedra to other complexes and classic number arrays.
Oct
20
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
added 185 characters in body
Oct
20
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
more connections
Oct
20
comment Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
@Dag Oskar Madsen, I've noticed that you are familiar with Koszul duality. That raises a flag since generating series for quadratic operads that are Koszul duals are related as reciprocals (linked to permutohedra) or comp. inverses (associahedra again), I believe, e.g., the self-dual Assoc-Assoc and the dual Lie-Com. Anything interesting there you can say concerning associahedra?
Oct
19
revised Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Eliminated a potentially confusing phrase
Oct
19
asked Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Oct
18
comment Cyclotomic Polynomials in Combinatorics
A059260 (A239473) now has a combinatorial interpretation.--For p prime, these are the h-polynomials for the n-simplexes.--Confer A049019 to see how substituting an e.g.f. for x can be used to generate a partition polynomial for the faces of permutahedra.