On the largest part size of low‐rank combinatorial assemblies

We give a general framework for approximations to decomposable combinatorial structures suitable to the situation where the size n$$ n $$ and the number of components k$$ k $$ are specified. In particular for assemblies, this involves a Poisson process, which, with the appropriate choice of parameter, may be viewed as an extension of saddlepoint approximation. We illustrate the use of the Poisson process description for assemblies by analyzing the component structure when the rank, defined as r:=n−k$$ r:= n-k $$ , is small relative to the size n$$ n $$ . There is near‐universal behavior, in the sense that, apart from degenerate cases where the exponential generating function has radius of convergence zero, we have for t∈(0,∞)$$ t\in \left(0,\infty \right) $$ and ℓ=1,2,…$$ \ell =1,2,\dots $$ : when r≍nα$$ r\asymp {n}^{\alpha } $$ for fixed α∈(ℓℓ+1,ℓ+1ℓ+2)$$ \alpha \in \left(\frac{\ell }{\ell +1},\frac{\ell +1}{\ell +2}\right) $$ , the size L1$$ {L}_1 $$ of the largest component converges in probability to ℓ+2$$ \ell +2 $$ ; when r∼tnℓ/(ℓ+1)$$ r\sim t\kern0.3em {n}^{\ell /\left(\ell +1\right)} $$ , we have ℙ(L1∈{ℓ+1,ℓ+2})→1$$ \mathbb{P}\left({L}_1\in \left\{\ell +1,\ell +2\right\}\right)\to 1 $$ , with the choice governed by a Poisson limit distribution for the number of components of size ℓ+2$$ \ell +2 $$ . This was recently observed, for the case ℓ=1$$ \ell =1 $$ and the special cases of permutations and set partitions, using Chen–Stein approximations for the indicators of attacks and alignments in rook placements.