Beta-sets, the shift, and the e-runner abacus of a partition

Let $\lambda ⊢n$ be a partition with English Young diagram $[\lambda ]$, parts ${\lambda }_1\ge \cdots \ge {\lambda }_m>0$ and conjugate ${\lambda }^{\mathrm{\prime }}$ (Partitions, English diagrams, and conjugation). For $1\le i\le m$ put $b_i(\lambda ):={\lambda }_i+m-i=h_{i,1}(\lambda ),$ the hook length of the first node of row $i$ (Hook, arm, leg, and hook length of a box); since $h_{i,1}(\lambda )={\lambda }_i+{\lambda }_1^{\mathrm{\prime }}-i$ and $m={\lambda }_1^{\mathrm{\prime }}$, the two expressions agree. Then $X_{\lambda }:=\{b_1(\lambda )>b_2(\lambda )>\cdots >b_m(\lambda )\}$ is the set of first-column hook lengths of $\lambda$, a finite strictly decreasing set of positive integers.

A $\beta$-set is a finite set $X=\{h_1>\cdots >h_m\}\subseteq {\mathbb{N}}_0$ of nonnegative integers, written in strictly decreasing order. For $l\in {\mathbb{N}}_0$ the $l$-shift of $X$ is $X+l:=\{h+l:h\in X\}\cup \{l-1,l-2,\dots ,1,0\}$ (for $l=0$ this is $X$ itself, the second set being empty), and the partition of $X$ is $P(X):=(h_1-(m-1),h_2-(m-2),\dots ,h_m-0),$ with trailing zeros deleted; this is a weakly decreasing sequence of nonnegative integers ending in $h_m$, hence a partition of ${\sum }_i{h}_i-m(m-1)\mathrm{/}2$ after deleting its zeros. A $\beta$-set for $\lambda$ is a $\beta$-set $X$ with $P(X)=\lambda$.

Let $e\ge 2$. James's $e$-abacus has $e$ runners $0,1,\dots ,e-1$ drawn left to right, with rows labelled $a=0,1,2,\dots$ downward; the position in row $a$ of runner $i$ carries the label $ae+i$. An $e$-abacus configuration $A$ is a finite set of beads placed in distinct positions of this abacus; a position without a bead is a gap. The configuration $A_X$ of a $\beta$-set $X$ carries a bead in the positions of $X$, and conversely $X_A$ is the set of position labels of the beads of $A$; the two constructions are mutually inverse, so they biject finite $\beta$-sets with finite bead configurations. Finally $P(A):=P(X_A)$, and $A_{\lambda }:=A_{X_{\lambda }}$.

• $P(X_{\lambda })=\lambda$. For $1\le i\le m$ one has $h_{i,1}(\lambda )-({\lambda }_1^{\mathrm{\prime }}-i)=({\lambda }_i+{\lambda }_1^{\mathrm{\prime }}-i)-({\lambda }_1^{\mathrm{\prime }}-i)={\lambda }_i$, so $X_{\lambda }$ is a $\beta$-set for $\lambda$; the shift convention of the definition is exactly the one for which this holds with $l=0$.

• Shifts describe all $\beta$-sets of $\lambda$. Let $X=\{h_1>\cdots >h_m\}$ be a $\beta$-set and $t=\mathrm{\ell }(\lambda )$ the number of parts of $\lambda$. By the definition of $P$ and the deletion of trailing zeros, $P(X)=\lambda$ if and only if $m\ge t$, $h_j-(m-j)={\lambda }_j$ for $j\le t$ and $h_j-(m-j)=0$ for $t<j\le m$. This holds if and only if $m-t\in {\mathbb{N}}_0$ and $h_j=({\lambda }_j+t-j)+(m-t)=b_j(\lambda )+(m-t)$ for $j\le t$ while $\{h_{t+1},\dots ,h_m\}=\{m-t-1,\dots ,1,0\}$; that is, if and only if $X=X_{\lambda }+(m-t)$. Hence the $\beta$-sets for $\lambda$ are exactly the sets $X_{\lambda }+l$ with $l\in {\mathbb{N}}_0$, and $l$ is recovered as $l=m-t$; in particular $P(X+l)=P(X)$ for every $\beta$-set $X$ and every $l$, since the two sets have the same $P$. This is Lemma 4.2 of the lecture notes and the display preceding (5.21) of the manuscript; it is the sense in which the shift is a normalisation and not a change of partition.

• The runner labelling. Writing $h=ae+i$ with $a\in {\mathbb{N}}_0$ and $i\in \{0,1,\dots ,e-1\}$ gives a bijection ${\mathbb{N}}_0\to {\mathbb{N}}_0\times \{0,\dots ,e-1\}$, so every position of the abacus has exactly one label and $X\mapsto A_X$ is a bijection onto the finite bead configurations, with inverse $A\mapsto X_A$. The runner of a position $h$ is its residue $h<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>e$; the residue of a bead is the residue of its label. No choice principle is used anywhere in this definition.