First-row padding of a partition

Let $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$ be a partition with $\mathrm{\mid }\alpha \mathrm{\mid }={\alpha }_1+\cdots +{\alpha }_k$ (Partitions, English diagrams, and conjugation), and let $n\ge 0$ be an integer. Read $\mathrm{\mid }\mathrm{\varnothing }\mathrm{\mid }=0$ and ${\alpha }_1=0$ when $\alpha =\mathrm{\varnothing }$. Whenever

$n\ge \mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1,$

define the first-row padding of $\alpha$ to size $n$ to be $\alpha [n]:=(n-\mathrm{\mid }\alpha \mathrm{\mid },{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k)\mathrm{.}$

The threshold is exact.

Above the threshold. Suppose $n\ge \mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$. Then $n-\mathrm{\mid }\alpha \mathrm{\mid }\ge {\alpha }_1\ge 0$, so every entry of the displayed sequence is a nonnegative integer, the entries sum to $(n-\mathrm{\mid }\alpha \mathrm{\mid })+\mathrm{\mid }\alpha \mathrm{\mid }=n$, and the sequence is weakly decreasing because $n-\mathrm{\mid }\alpha \mathrm{\mid }\ge {\alpha }_1$ and $\alpha$ is weakly decreasing (Partitions, English diagrams, and conjugation). If $\alpha$ is nonempty then ${\alpha }_1\ge 1$, hence $n-\mathrm{\mid }\alpha \mathrm{\mid }\ge 1$ and the displayed sequence is a partition of $n$ with positive parts. If $\alpha =\mathrm{\varnothing }$ then the displayed sequence is $(n)$; this is a partition of $n$ for every $n\ge 1$, while for $n=0$ one takes $\mathrm{\varnothing }[0]:=\mathrm{\varnothing }$, the unique partition of $0$. Thus the padding is defined and is a partition of $n$ on the whole displayed range.

Below the threshold. If $\alpha$ is nonempty and $n<\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$, then $n-\mathrm{\mid }\alpha \mathrm{\mid }<{\alpha }_1$; if also $n\ge \mathrm{\mid }\alpha \mathrm{\mid }$ the leading entry is nonnegative but smaller than the second entry ${\alpha }_1$, and if $n<\mathrm{\mid }\alpha \mathrm{\mid }$ it is negative. In both cases the displayed sequence is not a partition, so $\alpha [n]$ is left undefined. The threshold $\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$ is therefore the exact size at which the padded family begins: for nonempty $\alpha$, at $n=\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$ the padding equals $({\alpha }_1,{\alpha }_1,{\alpha }_2,\dots )$, which is a partition with a repeated first part when $\alpha$ is nonempty.

Neutrality of the notation. This definition records a family of partitions; it asserts no constancy of any quantity along the family and no stability property. The same notation is used by the FI-module and representation-stability developments; no FI-module structure or generation degree is asserted here.