Rrerku

I would like to create a Dyck path in Latex with two additional features. First, I would like to number all the East step except(!) for the last one. Secondly, for each valley (that is, an East step that is followed by a North step), I would like to draw "lasers" which would be lines that are parallel to the diagonal and that stops once it reaches the Dyck path. This is similar, but not exactly the same as the "laser construction" in this paper. See e.g. Figure 6.

I already have some code to obtain a Dyck path.

documentclassarticle
usepackagetikz

newcommandNEpath[4]- (#1);
 draw[help lines] (#1) grid +(#2,#3);
 draw[dashed] (#1) -- +(#3,#3);
 coordinate (prev) at (#1);
 foreach dir in #4
 ifnumdir=0
 coordinate (dep) at (1,0);
 else
 coordinate (dep) at (0,1);
 fi
 draw[line width=2pt,-stealth] (prev) -- ++(dep) coordinate (prev);
 ;


begindocument
 begintikzpicture
 NEpath0,0661,1,0,1,1,0,0,0,1,0,1,0;
 endtikzpicture
enddocument

which produces the following picture.

Whereas I would like to obtain something like

Is it be possible to modify my existing code to obtain what I desire? If not, is there an alternative approach?

Just for fun, something that adds the numbers and laser lines automatically. The laser lines are drawn automatically according to your clarified prescription. The strategy is to look at the elements of the list that are ahead and check whether or not they fulfill certain criteria. The integer vtest tests if the current point is a "valley", in which case it is 10, and the other integers are constructed similarly.

documentclass[tikz,border=3.14mm]standalone
newcounterDyckHsteps
begindocument
tikzsetcount list/.code 2 args=foreach XX [count=YY] in #1
xdef#2YY,Dyck arrow/.style=ultra thick,-stealth,
laser/.style=draw=blue,
Dyck path/.style=count list=#1DyckSteps,
/utils/exec=setcounterDyckHsteps0,insert path=%
foreach XX [count=YY,remember=YY as LastY (initially 0)]in #1
ifnumXX=0
edge[Dyck arrow] ++(1,0) ++(1,0) coordinate(Dyck-YY)
ifnumYY<DyckSteps
(Dyck-LastY) -- (Dyck-YY) node[midway,above]stepcounterDyckHstepsnumbervalueDyckHsteps
fi
else
edge[Dyck arrow] ++(0,1) ++(0,1) coordinate(Dyck-YY)
fi
pgfextrapgfmathtruncatemacrovtest0pgfmathtruncatemacroftest0pgfmathtruncatemacrohtest0pgfmathtruncatemacroitest1
pgfmathtruncatemacroRestStepsDyckSteps-YY
ifnumYY>1
 ifnumRestSteps>1
 pgfmathtruncatemacroftest#1[YY+1]+#1[YY]*10 % should be 10
 pgfmathtruncatemacrovtest#1[YY-1]+10*#1[YY] % valley test
 fi
 ifnumRestSteps>3
 pgfmathtruncatemacrohtestpow(-1,#1[YY+3])+pow(-1,#1[YY+2])
 +pow(-1,#1[YY+1])+pow(-1,#1[YY])+ifthenelse(#1[YY-1]==1,11,0))
 fi
 ifnumRestSteps>5
 pgfmathtruncatemacroitestpow(-1,#1[YY+5])+
 pow(-1,#1[YY+4])+pow(-1,#1[YY+3])+pow(-1,#1[YY+2])
 +pow(-1,#1[YY+1])+pow(-1,#1[YY])+ifthenelse(#1[YY-1]==1,11,0)
 +ifthenelse(#1[YY-2]==1,11,0)
 fi
fi%typeoutYY:RestSteps:ftest,htest,itest,vtest

ifnumvtest=10 
 %(Dyck-YY) node[blue,fill,circle,inner sep=2pt](Dyck-YY)
 ifnumitest=0
 (Dyck-YY) edge[laser] ++(3,3) (Dyck-YY)
 fi
 ifnumhtest=1100
 (Dyck-YY) edge[laser] ++(-2,-2) (Dyck-YY)
 fi
 ifnumftest=10
 (Dyck-YY) edge[laser] ++(1,1) (Dyck-YY)
 fi
fi 

begintikzpicture
 draw (0,0) grid (6,6);
 draw (0,0) [Dyck path=1,1,0,1,0,0,1,1,0,1,0,0];
endtikzpicture
enddocument

First, congratulations for figuring out everything by yourself. I upvoted your answer.

This answer is a slight improvement of your answer in the position of the nodes (the numbers). I use option above to have a better space between the number and the line below it.

documentclassarticle
usepackagetikz

newcommandNEpath[4]- (#1);
 draw[help lines] (#1) grid +(#2,#3);
 draw[dashed] (#1) -- +(#3,#3);
 coordinate (prev) at (#1);
 foreach dir in #4
 ifnumdir=0
 coordinate (dep) at (1,0);
 else
 coordinate (dep) at (0,1);
 fi
 draw[line width=2pt,-stealth] (prev) -- ++(dep) coordinate (prev);
 ;


begindocument
 begintikzpicture
 NEpath0,0661,1,0,1,0,0,1,1,0,1,0,0;
 draw (1,2) -- +(1,1);
 draw (3,3.1) -- +(2.9,2.9);
 draw (4,5) -- +(1,1);
 node[above=2pt] at (0.5, 2) 1;
 node[above=2pt] at (1.5, 3) 2;
 node[above=2pt] at (2.5, 3) 3;
 node[above=2pt] at (3.5, 5) 4;
 node[above=2pt] at (4.5, 6) 5;
 endtikzpicture
enddocument

With the help from JouleV, I managed to do solve this but I am leaving this as an answer for people who might be interested.

documentclassarticle
usepackagetikz

newcommandNEpath[4]- (#1);
 draw[help lines] (#1) grid +(#2,#3);
 draw[dashed] (#1) -- +(#3,#3);
 coordinate (prev) at (#1);
 foreach dir in #4
 ifnumdir=0
 coordinate (dep) at (1,0);
 else
 coordinate (dep) at (0,1);
 fi
 draw[line width=2pt,-stealth] (prev) -- ++(dep) coordinate (prev);
 ;


begindocument
 begintikzpicture
 NEpath0,0661,1,0,1,0,0,1,1,0,1,0,0;
 draw (1,2) -- +(1,1);
 draw (3,3.1) -- +(2.9,2.9);
 draw (4,5) -- +(1,1);
 node at (0.5, 2.5) 1;
 node at (1.5, 3.5) 2;
 node at (2.5, 3.5) 3;
 node at (3.5, 5.5) 4;
 node at (4.5, 6.5) 5;
 endtikzpicture
enddocument

This produces the following the picture

Perhaps not the most pretty solution and quite mechanical but it works for my purposes. However, It would still be interesting to have a more general solution so that I would not need to draw lines and numbers manually.