Littlewood-Richardson (LR) coefficients and Kostka Numbers appear in representation theory and combinatorics related to $GL_n$. It is known that Kostka numbers can be represented as special Littlewood-Rischardson coefficient. In this paper, we show how one can represent LR coefficient as a signed sum of Kostka numbers, and use the formulation to give a polynomial time algorithm for the same, hence showing that they belong to the same class of decision problems. As a corollary, we will prove Steinberg's formula using Kostant's partition function.

翻译：Littlewood-Richardson (LR) 系数和 Kostka 数字出现在与$GL_n$有关的演示理论和组合法中。 众所周知, Kostka 数字可以作为特殊Littlewood- Rischardson 系数来表示。 在本文中, 我们展示了如何将LL系数作为Kostka 数字的签名总和来表示, 并使用该公式来给出相同数字的多元时间算法, 从而显示它们属于同一类决策问题。 作为必然结果, 我们将用 Kostant 的分割函数来证明 Steinberg 的公式 。