# The alternating block decomposition of iterated integrals, and cyclic insertion on multiple zeta values

## TL;DR

This paper introduces a new combinatorial framework called the alternating block decomposition to generalize cyclic insertion conjectures for multiple zeta values, providing evidence through motivic frameworks.

## Contribution

The paper proposes the generalized cyclic insertion conjecture using alternating block decomposition, unifying previous conjectures and identities in multiple zeta values.

## Key findings

- Both the original BBBL cyclic insertion conjecture and Hoffman's identity are special cases of the new conjecture.
- Using Brown's motivic MZV framework, a symmetrized version of the conjecture is proven to hold up to a rational factor.
- The work offers new combinatorial tools and evidence supporting the broader conjecture.

## Abstract

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ \zeta(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ \pi $. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 \zeta(3,3,\{2\}^m) - \zeta(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see that both the original BBBL cyclic insertion conjecture, and Hoffman's conjectural identity, are special cases of this 'generalised' cyclic insertion conjecture. By using Brown's motivic MZV framework, we establish that some symmetrised version of the generalised cyclic insertion conjecture always holds, up to a rational; this provides some evidence for the generalised conjecture.