There were lots of "Fun with $F_{un}$" (field with one element) recent years. One of the points is that it provides bridge between geometrical and combinatorial questions, i.e. in the limit case $q=1$ geometry disappears and combinatorial structure distills, on the other hand looking through $F_1$ glass on combinatorics one may try to lift it to geometric picture over $F_q$/any field.

My question is about giving examples of that kind.

Question: What are some combinatorial/probabilistic identities/concepts which have nice $F_{un}$ interpretation and might be lifted to nice geometric identities/concepts over $F_q$/any field ? Something similar to the example below:

Example:

(See Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?)

Combinatorial side:

Consider Vandermonde identity: $$ \binom{m + n}{k} =\sum_{j} \binom{m}{k - j} \binom{n}{j} $$

Probability side: Consider $n+m$ balls, $m$ white, $n$ black, what is the probability to choose $k-j$ white, $j$ black balls ? Asnwer is:
$$ \frac{ \binom{m}{k - j} \binom{n}{j} } { \binom{m + n}{k} } $$

And since sum of probabilities over $j$ gives 1 we have the Vandermonde identity above.

First $F_1/F_q$ interpretation (projective geometry)

Interpretation: According to $F_1$-wisdom we should think about the Grassmanian when we see
binomial coefficient:
$$\binom{n}{k} = \#( Gr(k,n, F_1) ) $$

Hence we might expect some geometric identity related to Grassmanians, and indeed there is motivic identity which is true over any field:
$$ [Gr(k,m+n)] = \sum_j [Gr(k-j,m)][Gr(j,n)] [\mathbb{A}^{j(m-k-j)}], $$ As Sasha explained here: Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

On the level of $F_q$ points enumeration it gives q-Vandermonde identity.

Second interpretation (linear algebra)

I am not sure that this interpretation is fully correct, but let me give it. One of the curious things about $F_1$ is that linear and projective geometry coincide over it. They should be different by $GL(F_q)$ but over $F_1$ it is just $1$ (at least I see like that).

Interpretation:

$$ \binom{n}{k} = \# ( \Lambda^k V^n(F_1)) = dim ( \Lambda^k V^n) $$

I mean that number of elements in vector space coincide with dimension over $F_1$.

Now the Vandermonde identity can be interpreted like that: consider $V = V^{m} \oplus V^n$

$$ \Lambda^k V = \oplus_j \Lambda^{k-j} V^m\otimes \Lambda^{j} V^{n} $$

that gives a lift from enumeration to isomorphism of objects (linear spaces) which holds true not only over $F_1$, but actually over any field.

Some similar example can be found here: Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

As a reincarnation of Gjergji Zaimi's question q-Catalan numbers from Grassmannians it is natural to ask a particular case of question 1:

Question 2: Can one give any $F_{un}/F_q$ interpretation/lift of any identity related to Catalan numbers ?

There are plenty of facts about the Catalan numbers, but it seems not obvious to interpret them geometrically. See comments under above mentioned question

• q-Catalan number itself is NOT number of points of any smooth projective variety over $F_q$.