Assemblies of Ordered Partitions

• Assemblies of ordered partitions are combinatorial structures defined as sequences of ordered partitions with increasing minima and sum constraints.
• They enable unified enumeration through generalized Lah numbers, Stirling numbers, and explicit generating functions that capture rich algebraic interconnections.
• These assemblies underpin the classification of Bott–Samelson-type Bott manifolds and support moment-cumulant inversion in non-commutative stochastic analysis.

Assemblies of ordered partitions are combinatorial structures arising in the enumeration and classification of objects in algebraic topology, algebraic combinatorics, and stochastic analysis. These assemblies represent sequences of ordered partitions of a finite set, subject to specific conditions on their composition, and they appear centrally in the paper of Bott manifolds of Bott–Samelson type, generalized Lah numbers, ordered cumulants, and related generating functions. The combinatorics of assemblies of ordered partitions unify several strands of enumeration involving ordered blocks, constraint-driven partition combinatorics, and poset-theoretic Möbius inversion.

1. Ordered Partitions and Assemblies: Definitions and Basic Properties

An ordered partition of a finite set $[m]=\{1,2,\dots,m\}$ is a sequence $\tau=(\tau^1,\dots,\tau^r)$ of mutually disjoint nonempty subsets, where the order of the blocks is significant. The underlying (unordered) partition is formed by the set $\{\tau^1,\dots,\tau^r\}$, while $U(\tau)=\tau^1\cup\cdots\cup\tau^r$ gives the support, and $\min(\tau)=\min(U(\tau))$ picks out the smallest index involved.

An assembly of ordered partitions with bound $n$ is a sequence $\sigma=(\sigma_1,\dots,\sigma_\ell)$, where each $\sigma_a$ is itself an ordered partition of $[m]$, satisfying:

• The subsets $U(\sigma_1),\dots,U(\sigma_\ell)$ form a partition of $[m]$.
• The minima increase strictly: $$\min(\sigma_1)<\min(\sigma_2)<\cdots<\min(\sigma_\ell)$$.
• The sum condition: $r_1+r_2+\cdots+r_\ell+(\ell-1)\leq n$, where $r_a$ is the number of blocks in $\sigma_a$. These are denoted $AOP(n,m)$ (Jeong et al., 12 Nov 2025).

Assemblies are structurally decomposable into their constituent ordered partitions, enabling recursive construction and enumeration. The notion of neighbors in an assembly defines which elements may be transposed or considered adjacent, critical in characterizing equivalence relations and isomorphism classes.

2. Enumeration, Generating Functions, and Generalized Lah Numbers

Assemblies of ordered partitions are enumerated by various generalizations of classical combinatorial numbers:

• If unordered partitions are decorated by block orderings, the count of ordered partitions of $[n]$ into $k$ blocks is $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind. The total number of ordered partitions (the Fubini–Bell, or ordered-Bell, numbers) is $F_n = \sum_{k=0}^n k! S(n,k)$ (Bonnier et al., 2019).
• For assemblies encoding additional constraints (such as fixed block sizes and distinguished elements), Jehanne–Rath–Soria introduce generalized Lah numbers $\lah{n}{k}_{S,r}$, counting partitions of $[n+r]$ into ordered blocks of sizes in $S$ such that the first $r$ distinguished elements occupy distinct blocks (Bényi et al., 2020).
• The exponential generating function for these numbers is

$\sum_{n\ge k} \lah{n}{k}_{S,r} \frac{x^n}{n!} = \frac{1}{k!} \left( \sum_{s\in S} x^s \right)^k \left( \sum_{s\in S} s x^{s-1} \right)^r.$

• The enumeration of BS-type Bott manifolds is governed by

$$\sum_{n,m\ge1} \frac{b(n,m)}{m!} x^m y^n = \frac{1}{y(1-y)} \left( \exp\left[ \frac{y}{2} \left( \frac{1}{1-y(e^x-1)} + y(e^x-1) - 1 \right) \right] - 1 \right),$$

where $b(n,m)=|AOP(n,m)/\sim|$, and $\sim$ is blockwise reversal (Jeong et al., 12 Nov 2025).

Example Table: Enumeration in Small Cases

$m$	Shape in $AOP(n,m)$	$|AOP(n,m)|$ (Count)	$b(n,m)=|AOP(n,m)/\sim|$
2	$(1\,2)$, $(1|2)$, $(2|1)$, $((1),(2))$	4	$3$ for $n\ge2$
3	$(1\,2\,3)$, $(1|2\,3)$, etc. (see (Jeong et al., 12 Nov 2025))	19 (complete list casewise)	$b(3,3)=10$

This demonstrates the combinatorial explosion and rich structure in assemblies as parameters grow.

3. Lattice Structure and Möbius Inversion

The set of ordered partitions $\mathrm{Orp}(P)$ of a poset $(P,\le)$ forms a lattice with respect to refinement: $a\le b$ if every block of $a$ is contained in a block of $b$. The meet is common refinement, and the join is the minimal coarser ordered partition. For $P$ an antichain, $\mathrm{Orp}(P)$ specializes to totally ordered set partitions with the classical Fubini–Bell sequence appearing in enumeration (Bonnier et al., 2019).

The Möbius function on this lattice supports inversion formulas that parallel the classical moment-cumulant duality, but in the context of ordered partitions. The inversion weights and antichain ancestry are computed according to block factorials and signed combinatorial sums, with explicit formulas for the Möbius function in terms of block-structures:

$\vartheta(a) := \sum_{b\in A(a)} (-1)^{|b|-1} b!$

where $A(a)$ is the antichain ancestry of $a$.

For assemblies, the inverse of the matrix of generalized Lah numbers is interpreted via the Möbius function of certain posets of pairs, relating asterisk-lists and ordinary lists whose cardinalities are determined by set $S$ and parameter $r$ (Bényi et al., 2020).

4. Assemblies and Bott Manifolds of Bott–Samelson Type

Assemblies of ordered partitions have a central role in the classification of Bott manifolds of Bott–Samelson type, which are smooth projective toric varieties arising as desingularizations of Schubert varieties.

Jeong–Kim–Lee establish that each sequence $\bm i\in [n]^m$ corresponds (via explicit maps) to an assembly $\alpha(\bm i)\in AOP(n,m)$ and a Bott matrix $B_{\bm i}$, with the key result (Jeong et al., 12 Nov 2025):

• The set of Bott manifolds of Bott–Samelson type of size $m$ and parameter $n$ is in bijection with $AOP(n,m)/\sim$, where $\sim$ is block-wise reversal.
• Two assemblies correspond to the same Bott manifold if and only if they are related by block reversals.

The generating function for $b(n,m)$ therefore also counts Bott–Samelson-type Bott manifolds up to the natural involution.

5. Isomorphism Classes and Admissible Transpositions

To distinguish isomorphism classes of the associated manifolds, an equivalence $\approx$ on $AOP(n,m)$ is defined by admissible transpositions and blockwise reversals. An admissible transposition $s_j=(j,j+1)$ is allowed for $\sigma$ if $j$ and $j+1$ are not neighbors in $\sigma$.

Two assemblies are in the same $\approx$-equivalence class if one may be obtained from the other by a sequence of admissible transpositions and block reversals. This yields the classification:

• Isomorphism classes of Bott–Samelson-type Bott manifolds are in bijection with $AOP(n,m)/\approx$.
• The necessary and sufficient combinatorial invariant is thus the $\approx$-class of the assembly.

The correspondence between toric isomorphisms of Bott towers and these combinatorial operations is rigorously verified (Jeong et al., 12 Nov 2025).

6. Moment-Cumulant Structure and Ordered Partition Species

Ordered partitions also form the foundation for moment–cumulant theory in the context of non-commutative stochastic analysis. In particular, signature moments and cumulants of stochastic processes are indexed by ordered partitions, with the Möbius function of the ordered partition lattice governing the inversion between moments and cumulants (Bonnier et al., 2019). The combinatorial species framework expresses the set of ordered partitions as the species composition $\mathrm{Seq}\circ\mathrm{Set}_+$, with the exponential generating function

$\OP(x) = \frac{1}{1 - (e^x - 1)} = \frac{1}{2 - e^x}.$

This combinatorial foundation directly yields recurrences for ordered-Bell numbers and supports deep analytic parallels with classical partition-based cumulants.

7. Generalizations, Poset Structures, and Unified Enumeration

The framework for assemblies of ordered partitions generalizes in several directions:

• Generalized Lah numbers recover classical enumeration (unsigned Lah, Stirling, Fubini–Bell) as degenerate cases.
• Poset-based interpretations allow for the combinatorial and algebraic paper of inversion and Möbius function calculations in ordered block settings.
• Explicit determinantal formulas and generating functions admit closed-form enumeration for a diverse array of ordered partition assemblies (e.g., doubly-ordered partitions, Fubini-type counts under restriction) (Bényi et al., 2020).

A plausible implication is that the combinatorial infrastructure of assemblies of ordered partitions will continue to facilitate advances wherever ordered block decompositions interface with algebraic or geometric classification problems, particularly in enumerative geometry, combinatorial Hopf algebras, and higher-cumulant theories.