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6h
revised Hirzebruch's motivation of the Todd class
Some algebra
7h
comment Hirzebruch's motivation of the Todd class
see also mathoverflow.net/questions/10630/…
7h
comment Why do Bernoulli numbers arise everywhere?
For the Todd-Chern-Hirzebruch connection see mathoverflow.net/questions/60478/…
7h
revised Hirzebruch's motivation of the Todd class
Elaborated
1d
comment Why do Bernoulli numbers arise everywhere?
Erratum: For the soliton, from A145271, $dz/dt=df(t)/dt=g(f(t))=g(z)$ here with $f(x,t)= O^{(-1)}_{\bar{B}}(x,t)=z$, so connects the derivatives of $g$ to the f-vectors of the simplicial duals of permutahedra (Eulerians are the h-vectors), not those of the simplices. A134264 gives the critical link between the Bernoullis and normalized, reverse f-vectors of the self-dual simplices $\bar{B}_n$. Also, another link related to Todd class: mathoverflow.net/questions/10630/… .
1d
answered Hirzebruch's motivation of the Todd class
1d
revised What does the generating function $x/(1 - e^{-x})$ count?
Another graphical/analytical method.
1d
comment Why do Bernoulli numbers arise everywhere?
Yep, thanks. I was aware of those, but through the sciencedirect suggested references I spotted a very important connection to solitons (and the Ricatti equation) I hadn't noted before. Thanks very much for leading me near that trail.
1d
revised Why do Bernoulli numbers arise everywhere?
Introduced connections of Bernoullis to solitons and Kdv eqn.
2d
comment What does the generating function $x/(1 - e^{-x})$ count?
See also Hodges and Sukumar, "Bernoulli, Euler, permutations, and quantum algebras" and "Quantum algebra and parity dependent spectra".
Nov
20
comment What does the generating function $x/(1 - e^{-x})$ count?
@Qiaochu, I'm sort of astonished that with your background in combinatorics that you didn't mention the connection to surjections and permutahedra, but then again you cursorily dismissed mathoverflow.net/questions/53384/…. (What goes around, ... .)
Nov
20
revised What does the generating function $x/(1 - e^{-x})$ count?
added 28 characters in body
Nov
20
revised What does the generating function $x/(1 - e^{-x})$ count?
added 227 characters in body
Nov
20
answered What does the generating function $x/(1 - e^{-x})$ count?
Nov
20
comment Why do Bernoulli numbers arise everywhere?
The Bernoullis result from a dance of the reciprocals across the permutahedra, and vice versa. See oeis.org/A049019 and oeis.org/A133314 for relation to surjections, matrix reps for reciprocation, and a weighted graphs interpretation. See Buchstaber and Panov's "Toric Topology" for connections of polytopes to topology.
Nov
20
comment Why do Bernoulli numbers arise everywhere?
How about for penance saying two Hail Marys and finding two new articles? My answer is way too long to add more, and I'd be interested in some more perspectives. (I'll delete my comments.)
Nov
20
comment An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
There is a very interesting reference in the newer article, not in the earlier one.
Nov
20
revised Why do Bernoulli numbers arise everywhere?
Elaborated
Nov
19
revised Why do Bernoulli numbers arise everywhere?
Corrected a formula, some coefficients. Added link.
Nov
19
revised Why do Bernoulli numbers arise everywhere?
corrected the index of a formula