Invariants in partition classes

With $ p\left(n, k\right) $ denote the numerical value of the number of partitions of the natural number $ n $ on exactly $ k $ parts. Form an arithmetic progression of $ k $ natural numbers with an arbitrary first value $ x_1 = p\left(j, k\right)$, and the difference $ d = m \cdot LCM\left(1, 2, \dots, k\right) $, where $ j$ and $ m $ an arbitrary natural numbers. Calculate all the values of $ \left\{p\left(x_i, k\right)\right\}_{i = 1, 2, \dots, k} $ and make the alternating sum with the appropriate binomial coefficients $ \sum_{i = 0}^{k-1}\left(-1\right)^i \binom{k-1}{i}p\left(j+i\cdot d, k\right). $ The last sum has a constant value equal to $ \left(-1\right)^{k-1}\frac{d^{k-1}}{k!} $, regardless of the first selected member $ x_1 $ of the arithmetic progression. We call this sum the first partition invariant, and it exists in all classes. In addition to these values there are a whole number of other invariant values, but they exist only in some classes, and so forth.