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Hi all,

Let be $G_n=(V_n,E_n)$ a finite graph, where $V_n= \{0,1,\ldots, n\} \times\{0,1,\ldots,n\}$

and $E_n\subset V_n\times V_n$ V_n^{(2)}$is the edge set joining of the nearest neighboors neighbors in the$\ell^1$norm, that is,$\ E_n=\{\ \{z,w\}\subset V_n; \ \sum_{i=1}^2 |z_i-w_i| =1 \ \}.$Fix a vertex$x=(x_1,x_2)\in G_n$such that$x_2>x_1$(up-diagonal). I would like to know if it is true the following inequality:$\sharp[m,p]_{x}\leq \sharp[p,m]_x$, whenever$p < m$where$[m,p]_{x}$is the set of all spanning subgraphs of$G_n$satisfying the following properties: 1- the spanning subgraph has$m$horizontal edges and$p$vertical edges; 2- the vertices$(0,0)$and$x=(x_1,x_2)$are in the same connected component, and$\sharp A$is the cardinality of$A$. In other words, if I have avaliable more vertical edges than horizontal ones is it true that I can find more configurations connecting$0$and$x$if$x$is up-diagonal than in case that the quanties of horizontal and vertical are inverted ? Thanks in advance for any idea or reference. Edition: I would like to add a simple reduction for this problem. Since we have$(n+1)^2$vertices in any spanning subgraph the problem is solved if$m+p\geq (n+1)^2-1$, since any subgraph with this quantity of edges has a tree as subgraph on it. Therefore the reflection$R$along the axis x=y sends$[m,p]_x$to$[p,m]_x$bijectively and the inequality is this case becomes an equality. The non-connected case still unanswered. Any help is very welcome. 3 added 489 characters in body Hi all, Let be$G_n=(V_n,E_n)$a finite graph, where$V_n= \{0,1,\ldots, n\} \times\{0,1,\ldots,n\}$and$E_n\subset V_n\times V_n$is the edge set joining the nearest neighboors in the$\ell^1$norm, that is,$\ E_n=\{\ \{z,w\}\subset V_n; \ \sum_{i=1}^2 |z_i-w_i| =1 \ \}.$Fix a vertex$x=(x_1,x_2)\in G_n$such that$x_2>x_1$(up-diagonal). I would like to know if it is true the following inequality:$\sharp[m,p]_{x}\leq \sharp[p,m]_x$, whenever$p < m$where$[m,p]_{x}$is the set of all spanning subgraphs of$G_n$satisfying the following properties: 1- the spanning subgraph has$m$horizontal edges and$p$vertical edges; 2- the vertices$(0,0)$and$x=(x_1,x_2)$are in the same connected component, and$\sharp A$is the cardinality of$A$. In other words, if I have avaliable more vertical edges than horizontal ones is it true that I can find more configurations connecting$0$and$x$if$x$is up-diagonal than in case that the quanties of horizontal and vertical are inverted ? Thanks in advance for any idea or reference. Edition: I would like to add a simple reduction for this problem. Since we have$(n+1)^2$vertices in any spanning subgraph the problem is solved if$m+p\geq (n+1)^2-1$, since any subgraph with this quantity of edges has a tree as subgraph on it. Therefore the reflection$R$along the axis x=y sends$[m,p]_x$to$[p,m]_x\$ bijectively and the inequality is this case becomes an equality.
The non-connected case still unanswered. Any help is very welcome.

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