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Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$. Let $J_L:= \sum\limits_{i=1}^{l}{| [j_i,1]|}$ and $M_L:= \sum\limits_{i=1}^{l}{| [0,m_i]|}$. Set $X_L$:= $l-|J_L-M_L|$.

Question 1: Is there an easy proof that $X_L >0 $ ? This would prove Frankl's conjecture for distributive lattices (which is already known).

 

Question 2: What are the distributive lattices with $X_L=l$? Their number starts for $n \geq 3$ with 1,2,1,3,2,7,4. Examples are Boolean lattices.

 

Question 3: Let $U_n:= \sum\limits_{L \in \mathcal{L}_n}^{}{|J_L-M_L|}$, where $\mathcal{L}_n$ is the set of all distributive lattices on $n$ elements. $U_n/2$ starts for $n \geq 3$ with 0,0,1,2,6,12,34 (could it be https://oeis.org/A088808 ?). What is $U_n$?

Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$. Let $J_L:= \sum\limits_{i=1}^{l}{| [j_i,1]|}$ and $M_L:= \sum\limits_{i=1}^{l}{| [0,m_i]|}$. Set $X_L$:= $l-|J_L-M_L|$.

Question 1: Is there an easy proof that $X_L >0 $ ? This would prove Frankl's conjecture for distributive lattices (which is already known).

Question 2: What are the distributive lattices with $X_L=l$? Their number starts for $n \geq 3$ with 1,2,1,3,2,7,4. Examples are Boolean lattices.

Question 3: Let $U_n:= \sum\limits_{L \in \mathcal{L}_n}^{}{|J_L-M_L|}$, where $\mathcal{L}_n$ is the set of all distributive lattices on $n$ elements. $U_n/2$ starts for $n \geq 3$ with 0,0,1,2,6,12,34 (could it be https://oeis.org/A088808 ?). What is $U_n$?

Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$. Let $J_L:= \sum\limits_{i=1}^{l}{| [j_i,1]|}$ and $M_L:= \sum\limits_{i=1}^{l}{| [0,m_i]|}$. Set $X_L$:= $l-|J_L-M_L|$.

Question 1: What are optimal upper and lower bounds for $X_L$ for distributive lattices on $n$ elements? Note that $X_L >0$ would imply Frankl's conjecture for distributive lattices (which is already known). It seems the optimal lower bound is 1 and the optimal upper bound is $n-1$.

Question 2: What are the distributive lattices with $X_L=l$? Their number starts for $n \geq 3$ with 1,2,1,3,2,7,4. Examples are Boolean lattices.

Question 3: Let $U_n:= \sum\limits_{L \in \mathcal{L}_n}^{}{|J_L-M_L|}$, where $\mathcal{L}_n$ is the set of all distributive lattices on $n$ elements. $U_n$ starts for $n \geq 3$ with 0,0,1,2,6,12,34 (could it be https://oeis.org/A088808 ?). What is $U_n$?

Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$. Let $J_L:= \sum\limits_{i=1}^{l}{| [j_i,1]|}$ and $M_L:= \sum\limits_{i=1}^{l}{| [0,m_i]|}$. Set $X_L$:= $l-|J_L-M_L|$.

Question 1: Is there an easy proof that $X_L >0 $ ? This would prove Frankl's conjecture for distributive lattices (which is already known).

Question 2: What are the distributive lattices with $X_L=l$? Their number starts for $n \geq 3$ with 1,2,1,3,2,7,4. Examples are Boolean lattices.

Question 3: Let $U_n:= \sum\limits_{L \in \mathcal{L}_n}^{}{|J_L-M_L|}$, where $\mathcal{L}_n$ is the set of all distributive lattices on $n$ elements. $U_n/2$ starts for $n \geq 3$ with 0,0,1,2,6,12,34 (could it be https://oeis.org/A088808 ?). What is $U_n$?

Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$. Let $J_L:= \sum\limits_{i=1}^{l}{| [j_i,1]|}$ and $M_L:= \sum\limits_{i=1}^{l}{| [0,m_i]|}$. Set $X_L$:= $l-|J_L-M_L|$.

Question 1: What are optimal upper and lower bounds for $X_L$ for distributive lattices on $n$ elements? Note that $X_L >0$ would imply Frankl's conjecture for distributive lattices (which is already known). It seems the optimal lower bound is 1 and the optimal upper bound is $n-1$.

Question 2: What are the distributive lattices with $X_L=l$? Their number starts for $n \geq 3$ with 1,2,1,3,2,7,4. Examples are Boolean lattices.

Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$. Let $J_L:= \sum\limits_{i=1}^{l}{| [j_i,1]|}$ and $M_L:= \sum\limits_{i=1}^{l}{| [0,m_i]|}$. Set $X_L$:= $l-|J_L-M_L|$.

Question 1: What are optimal upper and lower bounds for $X_L$ for distributive lattices on $n$ elements? Note that $X_L >0$ would imply Frankl's conjecture for distributive lattices (which is already known). It seems the optimal lower bound is 1 and the optimal upper bound is $n-1$.

Question 2: What are the distributive lattices with $X_L=l$? Their number starts for $n \geq 3$ with 1,2,1,3,2,7,4. Examples are Boolean lattices.

Question 3: Let $U_n:= \sum\limits_{L \in \mathcal{L}_n}^{}{|J_L-M_L|}$, where $\mathcal{L}_n$ is the set of all distributive lattices on $n$ elements. $U_n$ starts for $n \geq 3$ with 0,0,1,2,6,12,34 (could it be https://oeis.org/A088808 ?). What is $U_n$?