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Let $\lambda$ be a partition of size at least 2, and let $n>0$ be an integer. Prove that $$s_{\lambda/(1)}.h_{n}=\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{strip} }s_{\lambda^{+}/(1)}-\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n-1\ \text{strip} }s_{\lambda^{+}}$$

where $s_\lambda$ and $s_{\lambda/(1)}$ are Schur and skew-Schur functions and $h_n$ is complete homogeneous symmetric function.

My idea: I want to use following two lemmas proved in Stanley's book in chapter 7 pages 339 and 340.

(1) We have $$s_vs_n=\sum_{\lambda}s_{\lambda}$$

summed over all partitions $\lambda$ such that $\lambda/v$ is a horizontal strip of size $n$.

(2) We have $$s_{\lambda/n}=\sum_{v}s_{v}$$ where $v$ ranges over all partitions $v\subseteq \lambda$ for which $\lambda/v$ is a horizontal strip of size $n$.

So $$s_{\lambda/(1)}=\sum_{\lambda^{-}\subseteq \lambda\\ \lambda/\lambda^{-} \ \text{hor.} 1 \ \text{strip}}s_{\lambda^{-}}$$

Whence $$s_{\lambda/(1)} h_n=s_{\lambda/(1)} s_n=\sum_{\lambda^{-}\subseteq \lambda\\ \lambda/\lambda^{-} \ \text{hor.} 1 \ \text{strip}}s_{\lambda^{-}}s_n=\sum_{\lambda^{-}\subseteq \lambda\\ \lambda/\lambda^{-}\ \text{hor.} 1 \ \text{strip}}\left(\sum_{v/\lambda^{-} \ \text{hor.} \ n \ \text{strip}}s_v\right)=\sum_{v,\lambda^{-}\\ \lambda/\lambda^{-} \text{hor.} \ 1 \ \text{strip}\\ v/\lambda^{-} \ \text{hor.} \ n\ \text{strip}}s_v $$

On the other hand,

$$\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{strip} }s_{\lambda^{+}/(1)}=\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{strip} }\left(\sum_{\lambda^{+}/v' \text{hor.} \ 1 \ \text{strip}}s_{v'} \right)=\sum_{\lambda^+,v'\\\lambda^{+}/\lambda \ \text{hor.} \ n\ \text{strip}\\ \lambda^{+}/v' \ \text{hor.} \ 1\ \text{strip}}s_{v'}$$

So it suffices to show

$$\sum_{v,\lambda^{-}\\ \lambda/\lambda^{-} \text{hor.} \ 1 \ \text{strip}\\ v/\lambda^{-} \ \text{hor.} \ n\ \text{strip}}s_v+\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n-1\ \text{strip} }s_{\lambda^{+}}=\sum_{\lambda^+,v'\\\lambda^{+}/\lambda \ \text{hor.} \ n\ \text{strip}\\ \lambda^{+}/v' \ \text{hor.} \ 1\ \text{strip}}s_{v'}$$

The last equation is intuitive but I could not show that.

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