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Let $P$ be a finite connected poset with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $P$ is defined as the matrix $M_P=-C^{-1}C^T$.

Let $u_P$ be defined as the trace of $M_P^2$.

Question 1: Is it true that $u_P$ is an odd integer in case $P$ is a lattice? Does it have a nice interpretation in this case?

I can prove question 1 for distributive lattices.

Question 2: Is $u_P$ an odd integer in case $P$ is just a bounded poset?

Questions 1 and 2 have a positive answer for posets with at most 9 points. In case this is true for general bounded posets, there surely is a nice reason.

For a general connected poset $P$ $u_p$ can be zero.

Here is how to get a poset (with a global maximum but not bounded) with $u_p=0$ using Sage:

n=6 

posets=[P for P in Posets(n) if P.is_connected()]

U=[P for P in posets if ((P.coxeter_transformation())^2).trace()==0]

P=U[0]

display(P)

The opposite poset (with a global minimum but not bounded) also has $u_P=0$.

Let $P$ be a finite connected poset with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $P$ is defined as the matrix $M_P=-C^{-1}C^T$.

Let $u_P$ be defined as the trace of $M_P^2$.

Question 1: Is it true that $u_P$ is an odd integer in case $P$ is a lattice? Does it have a nice interpretation in this case?

I can prove question 1 for distributive lattices.

Question 2: Is $u_P$ an odd integer in case $P$ is just a bounded poset?

Questions 1 and 2 have a positive answer for posets with at most 9 points. In case this is true for general bounded posets, there surely is a nice reason.

For a general connected poset $P$ $u_p$ can be zero.

Here is how to get a poset (not bounded) with $u_p=0$ using Sage:

n=6 

posets=[P for P in Posets(n) if P.is_connected()]

U=[P for P in posets if ((P.coxeter_transformation())^2).trace()==0]

P=U[0]

display(P)

Let $P$ be a finite connected poset with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $P$ is defined as the matrix $M_P=-C^{-1}C^T$.

Let $u_P$ be defined as the trace of $M_P^2$.

Question 1: Is it true that $u_P$ is an odd integer in case $P$ is a lattice? Does it have a nice interpretation in this case?

I can prove question 1 for distributive lattices.

Question 2: Is $u_P$ an odd integer in case $P$ is just a bounded poset?

Questions 1 and 2 have a positive answer for posets with at most 9 points. In case this is true for general bounded posets, there surely is a nice reason.

For a general connected poset $P$ $u_p$ can be zero.

Here is how to get a poset (with a global maximum but not bounded) with $u_p=0$ using Sage (https://sagecell.sagemath.org/) :

n=6

posets=[P for P in Posets(n) if P.is_connected()]

U=[P for P in posets if ((P.coxeter_transformation())^2).trace()==0]

P=U[0]

display(P)

The opposite poset (with a global minimum but not bounded) also has $u_P=0$.

Let $P$ be a finite connected poset with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $P$ is defined as the matrix $M_P=-C^{-1}C^T$.

Let $u_P$ be defined as the trace of $M_P^2$.

Question 1: Is it true that $u_P$ is an odd integer in case $P$ is a lattice? Does it have a nice interpretation in this case?

I can prove question 1 for distributive lattices.

Question 2: Is $u_P$ an odd integer in case $P$ is just a bounded poset?

Questions 1 and 2 have a positive answer for posets with at most 9 points. In case this is true for general bounded posets, there surely is a nice reason.

For a general connected poset $P$ $u_p$ can be zero.

Here is how to get a poset (with a global maximum but not bounded) with $u_p=0$ using Sage:

n=6 

posets=[P for P in Posets(n) if P.is_connected()]

U=[P for P in posets if ((P.coxeter_transformation())^2).trace()==0]

P=U[0]

display(P)

The opposite poset (with a global minimum but not bounded) also has $u_P=0$.

Let $P$ be a finite connected poset with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $P$ is defined as the matrix $M_P=-C^{-1}C^T$.

Let $u_P$ be defined as the trace of $M_P^2$.

Question 1: Is it true that $u_P$ is an odd integer in case $P$ is a lattice? Does it have a nice interpretation in this case?

I can prove question 1 for distributive lattices.

Question 2: Is $u_P$ an odd integer in case $P$ is just a bounded poset?

Questions 1 and 2 have a positive answer for posets with at most 9 points. In case this is true for general bounded posets, there surely is a nice reason.

For a general connected poset $P$ $u_p$ can be zero.

Here is how to get a poset (with a global maximum but not bounded) with $u_p=0$ using Sage (https://sagecell.sagemath.org/) :

n=6 posets=[P for P in Posets(n) if P.is_connected()] U=[P for P in posets if ((P.coxeter_transformation())^2).trace()==0] P=U[0] display(P)

Let $P$ be a finite connected poset with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $P$ is defined as the matrix $M_P=-C^{-1}C^T$.

Let $u_P$ be defined as the trace of $M_P^2$.

Question 1: Is it true that $u_P$ is an odd integer in case $P$ is a lattice? Does it have a nice interpretation in this case?

I can prove question 1 for distributive lattices.

Question 2: Is $u_P$ an odd integer in case $P$ is just a bounded poset?

Questions 1 and 2 have a positive answer for posets with at most 9 points. In case this is true for general bounded posets, there surely is a nice reason.

For a general connected poset $P$ $u_p$ can be zero.

Here is how to get a poset (with a global maximum but not bounded) with $u_p=0$ using Sage (https://sagecell.sagemath.org/) :

n=6

posets=[P for P in Posets(n) if P.is_connected()]

U=[P for P in posets if ((P.coxeter_transformation())^2).trace()==0]

P=U[0]

display(P)

The opposite poset (with a global minimum but not bounded) also has $u_P=0$.