Identifier
- St001650: Dyck paths ⟶ ℤ
Values
[1,0] => 2
[1,0,1,0] => 3
[1,1,0,0] => 3
[1,0,1,0,1,0] => 4
[1,0,1,1,0,0] => 4
[1,1,0,0,1,0] => 4
[1,1,0,1,0,0] => 4
[1,1,1,0,0,0] => 4
[1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,1,0,0] => 5
[1,0,1,1,0,0,1,0] => 5
[1,0,1,1,0,1,0,0] => 5
[1,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,0,1,0] => 5
[1,1,0,0,1,1,0,0] => 5
[1,1,0,1,0,0,1,0] => 5
[1,1,0,1,0,1,0,0] => 6
[1,1,0,1,1,0,0,0] => 5
[1,1,1,0,0,0,1,0] => 5
[1,1,1,0,0,1,0,0] => 5
[1,1,1,0,1,0,0,0] => 5
[1,1,1,1,0,0,0,0] => 5
[1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,1,0,0] => 6
[1,0,1,0,1,1,0,0,1,0] => 6
[1,0,1,0,1,1,0,1,0,0] => 6
[1,0,1,0,1,1,1,0,0,0] => 6
[1,0,1,1,0,0,1,0,1,0] => 6
[1,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,1,0,0,1,0] => 6
[1,0,1,1,0,1,0,1,0,0] => 4
[1,0,1,1,0,1,1,0,0,0] => 6
[1,0,1,1,1,0,0,0,1,0] => 6
[1,0,1,1,1,0,0,1,0,0] => 6
[1,0,1,1,1,0,1,0,0,0] => 6
[1,0,1,1,1,1,0,0,0,0] => 6
[1,1,0,0,1,0,1,0,1,0] => 6
[1,1,0,0,1,0,1,1,0,0] => 6
[1,1,0,0,1,1,0,0,1,0] => 6
[1,1,0,0,1,1,0,1,0,0] => 6
[1,1,0,0,1,1,1,0,0,0] => 6
[1,1,0,1,0,0,1,0,1,0] => 6
[1,1,0,1,0,0,1,1,0,0] => 6
[1,1,0,1,0,1,0,0,1,0] => 4
[1,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,1,0,1,1,0,0,0] => 4
[1,1,0,1,1,0,0,0,1,0] => 6
[1,1,0,1,1,0,0,1,0,0] => 6
[1,1,0,1,1,0,1,0,0,0] => 3
[1,1,0,1,1,1,0,0,0,0] => 6
[1,1,1,0,0,0,1,0,1,0] => 6
[1,1,1,0,0,0,1,1,0,0] => 6
[1,1,1,0,0,1,0,0,1,0] => 6
[1,1,1,0,0,1,0,1,0,0] => 4
[1,1,1,0,0,1,1,0,0,0] => 6
[1,1,1,0,1,0,0,0,1,0] => 6
[1,1,1,0,1,0,0,1,0,0] => 3
[1,1,1,0,1,0,1,0,0,0] => 4
[1,1,1,0,1,1,0,0,0,0] => 6
[1,1,1,1,0,0,0,0,1,0] => 6
[1,1,1,1,0,0,0,1,0,0] => 6
[1,1,1,1,0,0,1,0,0,0] => 6
[1,1,1,1,0,1,0,0,0,0] => 6
[1,1,1,1,1,0,0,0,0,0] => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => 7
[1,0,1,0,1,0,1,0,1,1,0,0] => 7
[1,0,1,0,1,0,1,1,0,0,1,0] => 7
[1,0,1,0,1,0,1,1,0,1,0,0] => 7
[1,0,1,0,1,0,1,1,1,0,0,0] => 7
[1,0,1,0,1,1,0,0,1,0,1,0] => 7
[1,0,1,0,1,1,0,0,1,1,0,0] => 7
[1,0,1,0,1,1,0,1,0,0,1,0] => 7
[1,0,1,0,1,1,0,1,0,1,0,0] => 10
[1,0,1,0,1,1,0,1,1,0,0,0] => 7
[1,0,1,0,1,1,1,0,0,0,1,0] => 7
[1,0,1,0,1,1,1,0,0,1,0,0] => 7
[1,0,1,0,1,1,1,0,1,0,0,0] => 7
[1,0,1,0,1,1,1,1,0,0,0,0] => 7
[1,0,1,1,0,0,1,0,1,0,1,0] => 7
[1,0,1,1,0,0,1,0,1,1,0,0] => 7
[1,0,1,1,0,0,1,1,0,0,1,0] => 7
[1,0,1,1,0,0,1,1,0,1,0,0] => 7
[1,0,1,1,0,0,1,1,1,0,0,0] => 7
[1,0,1,1,0,1,0,0,1,0,1,0] => 7
[1,0,1,1,0,1,0,0,1,1,0,0] => 7
[1,0,1,1,0,1,0,1,0,0,1,0] => 10
[1,0,1,1,0,1,0,1,0,1,0,0] => 12
[1,0,1,1,0,1,0,1,1,0,0,0] => 10
[1,0,1,1,0,1,1,0,0,0,1,0] => 7
[1,0,1,1,0,1,1,0,0,1,0,0] => 7
[1,0,1,1,0,1,1,0,1,0,0,0] => 12
[1,0,1,1,0,1,1,1,0,0,0,0] => 7
[1,0,1,1,1,0,0,0,1,0,1,0] => 7
[1,0,1,1,1,0,0,0,1,1,0,0] => 7
[1,0,1,1,1,0,0,1,0,0,1,0] => 7
[1,0,1,1,1,0,0,1,0,1,0,0] => 10
[1,0,1,1,1,0,0,1,1,0,0,0] => 7
[1,0,1,1,1,0,1,0,0,0,1,0] => 7
[1,0,1,1,1,0,1,0,0,1,0,0] => 12
[1,0,1,1,1,0,1,0,1,0,0,0] => 10
[1,0,1,1,1,0,1,1,0,0,0,0] => 7
>>> Load all 196 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path.
References
[1] Ringel, C. M. The finitistic dimension of a Nakayama algebra arXiv:2008.10044
Code
DeclareOperation("coordinateofsimple", [IsList]);
InstallMethod(coordinateofsimple, "for a representation of a quiver", [IsList],0,function(L)
local A,S,simA,n,U,UU;
A:=L[1];
S:=L[2];
simA:=SimpleModules(A);
n:=Size(simA);
U:=DimensionVector(S);
UU:=Filtered([1..n],x->U[x]>0);
return(Minimum(UU));
end
);
DeclareOperation("eofsimplemodule", [IsList]);
InstallMethod(eofsimplemodule, "for a representation of a quiver", [IsList],0,function(L)
local S,t1,U,t2;
S:=L[1];
t1:=ProjDimensionOfModule(S,33);
U:=Range(InjectiveEnvelope(S));
t2:=ProjDimensionOfModule(U,33);
return(Minimum(t1,t2));
end
);
DeclareOperation("Ringelbijectionforgivensimple", [IsList]);
InstallMethod(Ringelbijectionforgivensimple, "for a representation of a quiver", [IsList],0,function(L)
local S,t1,U,t2,e;
S:=L[1];
e:=eofsimplemodule([S]);
U:=Range(InjectiveEnvelope(S));
if IsOddInt(e)=true then return(NthSyzygy(S,e));
else return(NthSyzygy(U,e));fi;
end
);
DeclareOperation("Ringelbijection",[IsList]);
InstallMethod(Ringelbijection, "for a representation of a quiver", [IsList],0,function(LIST)
local A,simA,n,W,i,WW,WW2;
A:=LIST[1];
simA:=SimpleModules(A);
n:=Size(simA);
W:=[];for i in simA do Append(W,[Ringelbijectionforgivensimple([i])]);od;
WW:=[];for i in [1..n] do Append(WW,[[i,coordinateofsimple([A,W[i]])]]);od;
WW2:=[];for i in [1..n] do Append(WW2,[WW[i][2]]);od;
return([WW,Size(Set(WW2))=n]);
#first entry is map, second entry is whether it is a bijection.
end);
DeclareOperation("Ringelpermutationnormalform",[IsList]);
InstallMethod(Ringelpermutationnormalform, "for a representation of a quiver", [IsList],0,function(LIST)
local n,l,i,W,k,WW,WW2,O,f;
A:=LIST[1];
f:=Ringelbijection([A])[1];
W:=[];for i in f do Append(W,[i[2]]);od;
return(W);
end);
DeclareOperation("ringelmapaspermutation",[IsList]);
InstallMethod(ringelmapaspermutation, "for a representation of a quiver", [IsList],0,function(LIST)
local A,W,n,U,D;
A:=LIST[1];
W:=Ringelpermutationnormalform([A]);n:=Size(W);U:=PermList(W);
return(U);
end);
#example: A:=NakayamaAlgebra([3,3,2,1],GF(3));W:=Ringelpermutationnormalform([A]);
#z:=5;R:=[];for i in [2..z] do Append(R,[BuildSequencesLNak(i)]);od;R:=Union(R);RR:=[];for i in R do Append(RR,[[i,Ringelpermutationnormalform([NakayamaAlgebra(i,GF(3))])]]);od;RR;
DeclareOperation("OrderRingelpermutation",[IsList]);
InstallMethod(OrderRingelpermutation, "for a representation of a quiver", [IsList],0,function(LIST)
local n,l,i,W,k,WW,WW2,O,f;
A:=LIST[1];
f:=ringelmapaspermutation([A]);
return(Order(f));
end);
#example: A:=NakayamaAlgebra([3,3,2,1],GF(3));W:=OrderRingelpermutation([A]);
Created
Nov 29, 2020 at 19:57 by Rene Marczinzik
Updated
Nov 29, 2020 at 19:57 by Rene Marczinzik
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!