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I will introduce my question with the following example:

For the partition of 26 = 1+1+1+2+3+3+5+5+5 let us calculate its "fancy number" as follows:

From the group of the ones, we get: 1*(1+1)(1+1+1)=6 From the group of the twos (actually only one), we get: 2 From the group of the thress, we get: 3(3+3)=18 From the group of the fives, we get: 5*(5+5)*(5+5+5)=750

That is, for each distinct part we calculate the product of the cumulative sums obtained with that part.

Then, the fancy number of the partition 1+1+1+2+3+3+5+5+5 is the product of the above numbers: 6*2*18*750 = 162000

Another example: for the partitions of 6, the fancy numbers (fn) are:

for 1+1+1+1+1+1 fn is 1*2*3*4*5*6=720 for 1+1+1+1+2 fn is 1*2*3*4*2=48 for 1+1+1+3 fn is 1*2*3*3=18 for 1+1+2+2 fn is 1*2*2*4=16 for 1+1+4 fn is 1*2*4=8 for 1+2+3 fn is 1*2*3=6 for 2+2+2 fn is 2*4*6=48 for 2+4 fn is 2*4=8 for 3+3 fn is 3*6=18 for 1+5 fn is 1*5=5 for 6 fn is 6

which shows that two partitions can have the same fancy number. Now, my question is:

Do the "fancy numbers" recieve any special name? Have they been studied? What is it known about them?

thank you very much!

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1 Answer 1

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First of all, let's formalize the definition: Let $\lambda$ be a partition. For every integer $i \geq 1$, let $m_i\left(\lambda\right)$ be the number of appearances of $i$ in $\lambda$. Then your fancy number is $\prod\limits_{i\geq 1} \left(m_i\left(\lambda\right)! i^{m_i\left(\lambda\right)}\right)$. This number is a rather well-known number, called $z_{\lambda}$ in at least some books on this subject. It has the property that if $n$ is the weight of $\lambda$, and $\sigma$ is a permutation of $\left\lbrace 1,2,...,n\right\rbrace$ which has cycle type $\lambda$, then there are exactly $z_{\lambda}$ permutations of $\left\lbrace 1,2,...,n\right\rbrace$ which commute with $\sigma$. As a consequence, there are exactly $\dfrac{n!}{z_{\lambda}}$ distinct permutations with cycle type $\lambda$ in $S_n$.

Now that we know that these $z_{\lambda}$ figure in combinatorics, it is not surprising they appear in many formulas with symmetric functions, and thus in representation theory, Hopf algebra theory, $\lambda$-ring theory, algebraic topology, probably stochastics, particle physics, etc. For instance, there is the symmetric function identity

$\prod\limits_{i,j}\dfrac{1}{1-\xi_i\eta_j} = \sum\limits_{\lambda\text{ partition}} \dfrac{1}{z_{\lambda}} p_{\lambda}\left(\xi\right)p_{\lambda}\left(\eta\right)$,

where $\left(\xi_i\right)_i$ and $\left(\eta_j\right)_j$ are two disjoint sets of indeterminates (all pairwise commuting), and $p_{\lambda}$ denotes the $\lambda$-th power sum symmetric functions. For details, see all symmetric functions literature, e. g.,

I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition,

where they start appearing in §1.2 (in the context of the identity mentioned above) and continue to reappear in various context (counting equal terms in Pólya enumeration, Bell polynomials, etc.). If you look at the symmetric polynomials over $\mathbb Q$, you notice that they have a lot of bases; the $z_{\lambda}$ are the diagonal elements of the (triangular) transition matrix from the power-sum basis to the elementary-symmetric-polynomial basis.

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