Question

We know that Integer Partitions form a poset (actually we can define more than one partial orders on it), and so we can have some kind of Mobius function on it and consequently Mobius Inversion Formula (correct me if I am wrong)..... I am looking forward to some examples of applications of such kind of formula if it exists, and whether we can look Hook length formula as a manifestation of it or not.... Either you can give an example or point out some literature to look into....

Answer 1

The Möbius function of the poset $P_n$ of partitions of the integer $n$ ordered by refinement is not well-behaved, as is discussed e.g. in:

Günter Ziegler, On the poset of partitions of an integer, J. Combin. Theory Ser. A 42 (1986), no. 2, 215--222.

where the poset is shown not to be Cohen-Macaulay for $n\ge 19$ and not to have Möbius function on intervals that alternates in sign for $n\ge 111$. Thus, its order complex (nerve) on intervals is not always homotopy equivalent to a wedge of top dimensional spheres. Wanting nonetheless to understand the topology of the order complex for this poset and more general posets of multiset partitions was the original motivation for my work with Eric Babson on discrete Morse theory for posets, but we only obtained partial results in this direction, focusing mainly on $\mu_{P_n} (\hat{0},\hat{1})$ rather than arbitrary intervals.

If you are (more likely) interested in partially ordering the number partitions by shape containment rather than by refinement, then this is Young's lattice, and it is indeed well-behaved, as described e.g. at the wikipedia article on "Young's lattice". Young's lattice is a distributive lattice, hence each interval is shellable, and each interval has Möbius function equaling $0, 1, $ or $-1$. Specifically, it is 0 except for $\mu (\lambda_1,\lambda_2 )$ where the skew shape $\lambda_2 \setminus \lambda_1$ consists of single boxes touching at most at corners.

As far as applications, I would be looking in the theory of symmetric functions and in representation theory. There seems to be some discussion of this and references in the wikipedia article on "Young's lattice".

Answer 2

If you mean the poset you get with the relation "refinement" then its Möbius function is not known (see Exercise 122 in Chapter 3 of Stanley's Enumerative Combinatorics, 2nd edition).