Littlewood--Richardson tableaux and coefficients

Let $\lambda ,\nu$ be partitions with $[\lambda ]\subseteq [\nu ]$, and let $\mu$ be a partition with $\mathrm{\mid }\nu \mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\mu \mathrm{\mid }$. Use the conventions of Skew diagrams and semistandard skew tableaux for the skew diagram $\nu \mathrm{/}\lambda$ and for semistandard skew tableaux of shape $\nu \mathrm{/}\lambda$, and those of Semistandard tableaux and Kostka numbers for the content of a tableau; recall that entries of a semistandard skew tableau weakly increase along rows and strictly increase down columns (Partitions, English diagrams, and conjugation fixes the English row and column coordinates).

The reading word $w(T)$ of a semistandard skew tableau $T$ of shape $\nu \mathrm{/}\lambda$ is the word obtained by reading the rows of $T$ from right to left, beginning with the top row and proceeding to the bottom row. The word $w(T)=a_1{a}_2\cdots a_N$ is a lattice word (a lattice permutation) if in every prefix $a_1\cdots a_p$ and for every $i\ge 1$ the number of letters $i$ in the prefix is at least the number of letters $i+1$; the empty word is a lattice word vacuously.

A Littlewood--Richardson tableau (LR tableau) of shape $\nu \mathrm{/}\lambda$ and content $\mu$ is a semistandard skew tableau of shape $\nu \mathrm{/}\lambda$ and content $\mu$ whose reading word is a lattice word. The Littlewood--Richardson coefficient $c_{\lambda \mu }^{\nu }\in {\mathbb{Z}}_{\ge 0}$ is the number of LR tableaux of shape $\nu \mathrm{/}\lambda$ and content $\mu$; it is $0$ when $[\lambda ]\subseteq ̸[\nu ]$, when $\mathrm{\mid }\nu \mathrm{\mid }\mathrm{\ne }\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\mu \mathrm{\mid }$, or when no such tableau exists. For $\mu =\mathrm{\varnothing }$ and $\nu =\lambda$ the empty skew tableau is the unique tableau of content $\mathrm{\varnothing }$, its reading word is empty, and hence $c_{\lambda \mathrm{\varnothing }}^{\lambda }=1$; more generally $c_{\lambda \mu }^{\lambda }=0$ unless $\mu =\mathrm{\varnothing }$, and $c_{\lambda \mu }^{\nu }=0$ unless $\lambda \subseteq \nu$ and $\mathrm{\mid }\nu \mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\mu \mathrm{\mid }$.