# Littlewood-Richardson sequences and Littlewood-Richardson coefficients

I'm looking for a proof or a reference for the following statement, I give the definitions below:

There exists a Littlewood-Richardson sequence of type $(\alpha, \beta, \lambda)$ if and only if $c_{\alpha, \beta}^{\lambda}\not = 0$.

Let $A = [\alpha^0, \alpha^1, \cdots, \alpha^r]$ be a sequence of partitions. $A$ is Littlewood-Richardson (or a L-R sequence) if

1) for $h=1,2,\cdots , r$ we have $0 \leq \alpha_i^h - \alpha_i^{h-1} \leq 1$ for every $i$.

2) for $h=2, \cdots, r$ we have $\sum_{i \geq k} (\alpha_i^h - \alpha_i^{h-1}) \leq \sum_{i \geq k} (\alpha_i^{h-1} - \alpha_i^{h-2})$ for every $k$.

if $\alpha = \alpha^0$, $\alpha^r = \lambda$, $|\alpha^h| - |\alpha^{h-1}| = m_h$, and $\overline{\beta} = (m_1, m_2, \cdots, m_r, 0, \cdots)$ we call $A$ a L-R sequence of type $(\alpha, \beta, \lambda)$.

($\overline{\beta}$ being the conjugate partition to $\beta$).

Thanks!

• Your definition of LR-sequence might be translatable to the notion of Yamanouchi-tableau of shape $\lambda/\alpha$, and weight given by $\beta$... May 18 '17 at 6:18
• I don't have time to work out the details, but I suspect it's more like Yamanouchi tableaux of shape $\lambda^t/\alpha^t$ and weight given by $\beta^t$. The thing is, condition 1 says that $\alpha^{h-1} / \alpha^h$ is a vertical strip. May 18 '17 at 9:15
• T. Klein, The Hall Polynomial, Journal of Algebra Volume 12, Issue 1, May 1969, Pages 61-78 explains a bijection to Yamanouchi tableaux on its first two pages. May 18 '17 at 9:18