Finite-Size Corrections to the Boltzmann–Shannon Correspondence for Set Partitions with Broken Block-Size Symmetry

We study the finite-N discrepancy between the combinatorial logarithm of a partition multiplicity and its leading Shannon-entropy approximation. Starting from the exact symmetry factor for pure (unlabeled) set partitions, we derive the full second- and third-order Stirling expansion of lnΩ for a labeled occupation vector n=(n1,…,nk), recovering the known sub-extensive logarithmic correction −(k−1)/2 ln N together with an explicit O(N−1) inverse-volume term. We then combine this expansion with the block-size degeneracy factor of a pure partition to obtain a master formula in which the total combinatorial entropy splits into an extensive occupation term NH(p) and a sub-extensive block-degeneracy term kH_sym(m). Numerical verification on finite systems (N≤160) confirms the predicted scaling of the successive truncation errors: O(ln N), O(N−1), and O(N−3). The precise finite-size corrections obtained here are relevant for small-system statistical mechanics and for bias correction in entropy estimation. For completeness we also record, as a classical result, the asymptotic S(N,k) ∼ k^N/k! for Stirling numbers of the second kind at fixed k, and explain why it is governed by an exponentially small correction rather than the power-law correction obtained for the labeled multinomial.