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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 $2$.

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  • $\begingroup$ Sorry, how are these coefficients defined? I am familiar with LR coefficients depending on three diagrams. $\endgroup$
    – Fedor Petrov
    May 19 at 15:55
  • $\begingroup$ 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. $\endgroup$
    – John Murray
    May 19 at 16:07
  • $\begingroup$ So, you write $C(\gamma/\alpha, \beta)$ for $C_{\alpha, \beta}^\gamma$, right? $\endgroup$
    – Fedor Petrov
    May 19 at 18:49
  • $\begingroup$ Yes. I guess that hides the commutativity of the coefficients! $\endgroup$
    – John Murray
    May 19 at 20:37
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    $\begingroup$ @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. $\endgroup$
    – John Murray
    May 20 at 14:33

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