$\newcommand{\Fun}{F_\mathrm{un}}$There were lots of "Fun with $\Fun$" (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.

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

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  ? Answer 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 Grassmannians, 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.

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).

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

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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

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 $F_q$.

$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in 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 a $\Fun$ glass on combinatorics one may try to lift it to a geometric picture over $F_q$/any field.

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

Consider the 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? The answer is:
$$ \frac{ \binom{m}{k - j} \binom{n}{j} } { \binom{m + n}{k} }. $$

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

First $\Fun/F_q$ interpretation (projective geometry)

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

Hence we might expect some geometric identity related to Grassmannians, 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.

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

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

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

I mean that number of elements in a vector space coincide with its dimension over $\Fun$.

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

$$ {\bigwedge}^k V = \bigoplus_j {\bigwedge}^{k-j} V^m\otimes {\bigwedge}^{j} V^{n} $$

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

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

There are plenty of facts about the Catalan numbers, but it seems not obvious to interpret them geometrically. See Will Sawin's comment under the above mentioned question — the $q$-Catalan number itself is NOT the number of points of any smooth projective variety over $F_q$.

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 $F_q$.

$\newcommand{\Fun}{F_\mathrm{un}}$There were lots of "Fun with $\Fun$" (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 $\Fun$ 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 ? Answer 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 Grassmannians, 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 $\Fun/F_q$ 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 $F_q$.

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 dissappears and combinatorial structures distills, on the other hand looking throw $F_1$ 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/probabilitic 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 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: $$ \binom{m + n}{k} =\sum_{j} \binom{m}{k - j} \binom{n}{j} $$

Probality 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 Vandermode 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 exaplained 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 interpreation is fully correct, but let me give it. One of 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 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 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 $F_q$.

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 $F_q$.