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EXAMPLES OF RECENT RESULTS (or Bijections in my case!)
One funny and interesting advantage of doing research in enumerative combinatorics, is that one can easily state many of the results by casually saying: “I have counted the following objects or structures!”. That’s exactly what I will try to list below:
The number of connected chord diagrams avoiding bottom cycles is given by the generalized Catalan number
There is a bijection between connected chord diagrams avoiding cycles (top and bottom) and the intervals of the Kreweras lattice of noncrossing partitions.
Sequence A088221 of the OEIS counts chord diagrams with at most two connected components.
2-Connected chord diagrams C≥2(x) are functionally related to connected chord diagrams C(x) as per
Applying the suitable ‘alien derivative’ to the factorially divergent sequences in the previous equation, and using Lagrange Inversion, we found that .
By the above result we have built the machinery required to get all coefficients, some of which were conjectured in previous works. For example, the first 5 coefficients in the above expansion were conjectured in the work of D. Broadhurst on Quantum Electrodynamics graphs. It also generalizes the result by Kleitman on linked diagrams.
An interesting consequence discovered accordingly was that, renormalization counterterms (terms subtracted from divergent Feynman integrals in a subtraction renormalization scheme in order to extract the useful information of the field observed) for Quenched QED (that’s simply QED with no fermion loops in the Feynman diagrams) is functionally related to the number of 2-connected chord diagrams. This offered a simple way to study the asymptotics of these counterterms without the need for singularity analysis, which has been the classic procedure for physicists. This result was extensively related to the work of M. Borinsky on perturbation theory.
I have also proved that the number of Yukawa 1PI tadpole graphs with loop number n is equal to the number of connected chord diagrams on n chords. This was done by introducing a reversible algorithm that transforms Yukawa graphs into chord diagrams. For example, on the left we have the 27 1PI Yukawa tadpole graphs, consisting of boson-fermion interactions and allowing 4 loops. Each of these Feynman diagrams can be decrypted into a specific one of the 27 connected chord diagrams with 4 edges!
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