---

Fix a positive integer $m$. A template is a nonempty nondecreasing word $T\in[m]^\ell$. Let $\operatorname{Perm}(T)$ be its distinct rearrangements. Choose disjoint nonempty sets $I_1,\ldots,I_\ell\subseteq[n]$ of a common size $d\ge1$, and fix one letter at every coordinate outside their union. For each $v\in\operatorname{Perm}(T)$, put $v_j$ throughout $I_j$. The resulting collection is a uniform block set with template $T$. Its block size, called its degree in this paper, is $d$.

If letter $a$ has multiplicity $r_a$ in $T$, the collection has $\ell!/\prod_a r_a!$ distinct words. Nonempty blocks ensure that distinct rearrangements give distinct words. In particular, template $11223$ has $30$ words, whereas $123$ has six.

The pattern records the labels of the active blocks as their coordinates are read increasingly, omitting all fixed coordinates. Thus pattern $ABCCBA$ means three two-element blocks occupying ranks $\{1,6\}$, $\{2,5\}$ and $\{3,4\}$ among the six active coordinates. The active coordinates need not be consecutive in $[n]$.

The lower-bound proof also permits unequal block sizes, provided each lies between $1$ and the stated bound. This stronger obstruction includes uniform block sets. Allowing empty blocks would destroy that assertion.

Convention change. In Leader–Russell–Walters, degree means the total number of active coordinates, namely $\ell d$ for a $d$-uniform block set. Their general definition also permits empty blocks. Neither convention can be substituted silently for the present one.

Source. Published paper, pp. 2 and 5, and arXiv v1, pp. 2 and 5. See the source digest for the two versions and version qualifications.

Bears on. #174.