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Block-transitive designs with a poset of imprimitive partitions

Carmen Amarra, Alice Devillers, Cheryl E. Praeger

TL;DR

Block-transitive designs that preserve a poset of partitions are studied with automorphism groups modeled by generalised wreath products. The authors derive a necessary-and-sufficient criterion, expressed via the array function of a base block, for the orbit of the base block under the generalized wreath product to form a 2-design, unifying prior chain/antichain results. They provide explicit infinite families of 2-designs for all posets of order 3 and for the N-poset, demonstrating the practicality of constructing highly symmetric designs with prescribed imprimitive structures. The approach combines ancestral-subset analysis, orbit counting, and NP-hard-looking combinatorics into a concrete design-construction framework with broad applicability in combinatorial design theory and statistical experimental design.

Abstract

We study block designs which admit an automorphism group that is transitive on blocks and points, and leaves invariant every partition in a given finite poset of partitions of the point set. The full stabiliser $G$ of all the partitions in the poset is a generalised wreath product. We use the theory of generalised wreath products to give necessary and sufficient conditions, in terms of the `array' of a point-subset $B$, for the set of $G$-images of $B$ to form the block-set of a $G$-block-transitive $2$-design. This generalises previous results for the special cases where the poset is a chain or an anti-chain. We also give explicit infinite families of examples of $2$-designs for each poset involving three proper partitions, and for the famous $N$-poset with four partitions. (Posets with two proper partitions have been treated previously.) This suggests the problem of finding explicit examples for other posets.

Block-transitive designs with a poset of imprimitive partitions

TL;DR

Block-transitive designs that preserve a poset of partitions are studied with automorphism groups modeled by generalised wreath products. The authors derive a necessary-and-sufficient criterion, expressed via the array function of a base block, for the orbit of the base block under the generalized wreath product to form a 2-design, unifying prior chain/antichain results. They provide explicit infinite families of 2-designs for all posets of order 3 and for the N-poset, demonstrating the practicality of constructing highly symmetric designs with prescribed imprimitive structures. The approach combines ancestral-subset analysis, orbit counting, and NP-hard-looking combinatorics into a concrete design-construction framework with broad applicability in combinatorial design theory and statistical experimental design.

Abstract

We study block designs which admit an automorphism group that is transitive on blocks and points, and leaves invariant every partition in a given finite poset of partitions of the point set. The full stabiliser of all the partitions in the poset is a generalised wreath product. We use the theory of generalised wreath products to give necessary and sufficient conditions, in terms of the `array' of a point-subset , for the set of -images of to form the block-set of a -block-transitive -design. This generalises previous results for the special cases where the poset is a chain or an anti-chain. We also give explicit infinite families of examples of -designs for each poset involving three proper partitions, and for the famous -poset with four partitions. (Posets with two proper partitions have been treated previously.) This suggests the problem of finding explicit examples for other posets.

Paper Structure

This paper contains 19 sections, 10 theorems, 87 equations, 8 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Previous results on chains and antichains
  3. Poset block structures and generalised wreath products of groups
  4. Posets and their ancestral subsets
  5. Poset block structures and partitions
  6. Generalised wreath products of groups
  7. Orbitals and orbits on unordered pairs
  8. Direct products of poset block structures
  9. Kronecker products of poset block structures
  10. Array function of a block
  11. Proof of Theorem \ref{['t:gwp-design']}
  12. Application to chains and antichains
  13. Families of examples for posets of small order
  14. Posets $(I,\preccurlyeq)$ with $|I| = 3$
  15. Example for Figure \ref{['fig:ch-grid']}
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $\mathscr{I}=(I,\preccurlyeq)$ be a finite partially ordered set with $|I|\geq 2$, and for each $i \in I$ let $\Delta_i$ be a finite set of size $e_i \geq 2$, and set ${\mathscr{P}} = \prod_{i \in I} \Delta_i$, of size $v = \prod_{i \in I} e_i$. For each $i\in I$ let $G_i \leq {\mathrm{Sym}}(\De where $J^{\rm c}$ denotes the complement of $J$ in $I$, and $\partial{J}$ is the set of all maximal

Figures (8)

  • Figure 1: Poset block structures for partially ordered sets $(I,\preccurlyeq)$ with $|I| = 2$
  • Figure 2: Poset block structures for partially ordered sets $(I,\preccurlyeq)$ with $|I| = 3$; chain and antichain
  • Figure 3: Poset block structures for partially ordered sets $(I,\preccurlyeq)$ with $|I| = 3$; not chain or antichain
  • Figure 4: Poset block structure for the 'N' poset
  • Figure 5: Distribution of points of $B$ in Example \ref{['ex:ch-grid']}
  • ...and 3 more figures

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • ...and 13 more