# Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?

Let $$n\in{\mathbb N}$$ and write $$n=q_1+q_2+\dots+q_t$$, where $$q_1>q_2>\dots>q_t$$ are powers of $$2$$. Let $$\lambda_n$$ be the partition with Frobenius symbol $$(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1},\dots,q_2,q_1)$$. Thus $$\lambda_n$$ is a partition of $$2n$$ whose diagonal hooks have lengths $$2q_1,2q_2,\dots,2q_t$$ and arm lengths $$q_1,q_2,\dots,q_t$$, respectively. For example $$\lambda(15)=(9,6,5,5,2,1,1,1)$$.

Next consider the triangular partition $$\delta_n:=(n,n-1,\dots,3,2,1)$$ and the near triangular partition $$\tau_n:=(n,n-1,\dots,3,2)$$. Notice that the skew-partition $$\tau_{n+1}/\delta_{n-1}$$ has staircase shape, with 2 boxes in each of its $$n$$ rows. Consider the Littlewood-Richardson coefficient $$C_n:=C_{\delta_{n-1},\lambda_n}^{\tau_{n+1}}$$ (using lrcalc in SageMath): $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} n&1&2&3&4&5&6&7&8&9&10&11\\ \hline C_n&1&1&1&1&9&49&89&1&49&1737&16121 \end{array}$$

$$\begin{array}{c|c|c|c|c|c} n&12&13&14&15&16\\ \hline C_n&281521&10414313&177901001&895807073&1\\ \end{array}$$

$$\begin{array}{c|c|c|c|c|c} n&17&18&19&20&21\\ \hline C_n&225&37681&1579025&105586569&22888140273 \end{array}$$

My main question is whether anyone can show or explain why $$C_n\equiv1(\hspace{-.3cm}\mod{8}),\quad\mbox{for all n>0?}$$

It is easy to show that $$C_n>0$$, for all $$n$$ (fill the first $$q_1$$ rows of the skew-diagram with the symbols in the first diagonal hook of a row-filling of $$\lambda_n$$, repeat with the other rows and diagonal hooks). Moreover $$C_n=1$$, when $$n$$ is a power of $$2$$ (as $$\lambda_n$$ is then a hook partition). Also notice that $$C_n=(n-2)^2$$, when $$n=q+1$$, for $$q=2,4,8,16$$. I do not understand this, but $$\lambda_n$$ is close to being a hook for these values of $$n$$.

Another question is to give upper and lower bounds for $$C_n$$, and show that $$C_n$$ is unbounded, as $$n\rightarrow\infty$$. Also show that $$C_n$$ is monotone increasing between $$q+1$$ and $$2q-1$$, for all $$q>2$$ a power of $$2$$.

• Sorry, how are these coefficients defined? I am familiar with LR coefficients depending on three diagrams. May 19 at 15:55
• For partitions $\alpha,\beta$ and $\gamma$, $C_{\alpha,\beta}^\gamma$ is the coefficient of the Schur function $s_\gamma$ in $s_\alpha s_\beta$. I'm guessing that your diagram is a Knutson-Tao hive, honeycomb or a Berstein-Zelevinsky triangle. A more traditional method (used here) is to consider Littlewood-Richardson tableau of shape $\gamma/\alpha$ and weight $\beta$ (or of shape $\gamma/\beta$ and weight $\alpha$). An L-R tableau is semi-standard (non-decreasing on rows, strictly increasing on columns) with a word which is lattice. All these methods are of course equivalent. May 19 at 16:07
• So, you write $C(\gamma/\alpha, \beta)$ for $C_{\alpha, \beta}^\gamma$, right? May 19 at 18:49
• Yes. I guess that hides the commutativity of the coefficients! May 19 at 20:37
• @Fedor Let $\lambda$ be a partition, with transpose $\lambda^t$. The diagonal hook in the $r$-th row of $\lambda$ has arm length $a=\lambda_r-r$ and leg length $\ell=\lambda^t_r-r$. Thus it contains $a+\ell+1$ boxes; the diagonal box plus the $a$ boxes in row $i$ strictly to the right and the $\ell$ boxes in column $i$ strictly below. I think this is standard terminology. May 20 at 14:33