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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 ToddChernHirzebruch 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 fvectors of the simplicial duals of permutahedra (Eulerians are the hvectors), not those of the simplices. A134264 gives the critical link between the Bernoullis and normalized, reverse fvectors of the selfdual 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

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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 