Padded Kronecker coefficient sequences

Let $\alpha ,\beta ,\gamma$ be partitions (Partitions, English diagrams, and conjugation) and put $n_0:=\max \{\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1,\mathrm{\mid }\beta \mathrm{\mid }+{\beta }_1,\mathrm{\mid }\gamma \mathrm{\mid }+{\gamma }_1\},$ where the first part of the empty partition is read as $0$. For every integer $n\ge n_0$ the three first-row paddings $\alpha [n]$, $\beta [n]$, $\gamma [n]$ are partitions of $n$, by First-row padding of a partition and the choice of $n_0$ as the maximum of the three thresholds. On that range define the size-indexed padded Kronecker coefficients by $g_n(\alpha ,\beta ,\gamma ):=g(\alpha [n],\beta [n],\gamma [n]),$ the Kronecker coefficient of three partitions of $n$ (Kronecker coefficients and the internal tensor product).

Domain. The threshold $n_0$ is exactly where the common domain of the three padded families starts: for $n\ge n_0$ all three padded sequences are partitions of $n$, while for $n<n_0$ at least one of the three padded sequences is not a partition of $n$ and the expression is left undefined. The padded Kronecker coefficient sequence, indexed from $0$ on $\mathbb{N}$, is $k⟼g_{n_0+k}(\alpha ,\beta ,\gamma )(k\in \mathbb{N})\mathrm{.}$ Every term is defined since $n_0+k\ge n_0$. The notation $g_n$ retains $n$ as the common size of the padded partitions: it is defined only for $n\ge n_0$, and no values at smaller sizes are introduced by reindexing.

No constancy is asserted. This definition introduces notation only. It asserts neither that $g_n(\alpha ,\beta ,\gamma )$ is eventually constant, nor any bound on it, nor any comparison between different values of $n$; Murnaghan stability is a theorem about this sequence proved elsewhere, and this definition is available as its noncircular input.