Explicit distribution of the number of degree-two vertices in HSC(2)

Derive an explicit general formula for the probability distribution Q(N_2,N) of the number of degree-two vertices in the HSC(2) model.

Background

For HSC(2), the paper determines the mean, variance, higher cumulant asymptotics, and several extreme probabilities for N_2, the number of degree-two vertices. It also derives an asymptotically Gaussian form through a continuum approximation.

An exact finite-size formula for the full distribution Q(N_2,N), however, is not obtained.

References

We have not found an explicit general formula for $Q(N_2,N)$. Some exact results, e.g., explicit formulae for the extreme probabilities

— Random Recursive Simplicial Complexes  (2608.26547 - Krapivsky et al., 27 Aug 2026) in Appendix, Section “Homogeneous simplicial complexes”