- FindStat

Identifier

St001244: Dyck paths ⟶ ℤ (values match St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.)

Values

[1,0] => 0

[1,0,1,0] => 1

[1,1,0,0] => 0

[1,0,1,0,1,0] => 1

[1,0,1,1,0,0] => 1

[1,1,0,0,1,0] => 1

[1,1,0,1,0,0] => 1

[1,1,1,0,0,0] => 0

[1,0,1,0,1,0,1,0] => 1

[1,0,1,0,1,1,0,0] => 1

[1,0,1,1,0,0,1,0] => 2

[1,0,1,1,0,1,0,0] => 1

[1,0,1,1,1,0,0,0] => 1

[1,1,0,0,1,0,1,0] => 1

[1,1,0,0,1,1,0,0] => 1

[1,1,0,1,0,0,1,0] => 1

[1,1,0,1,0,1,0,0] => 2

[1,1,0,1,1,0,0,0] => 1

[1,1,1,0,0,0,1,0] => 1

[1,1,1,0,0,1,0,0] => 1

[1,1,1,0,1,0,0,0] => 1

[1,1,1,1,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,0] => 1

[1,0,1,0,1,0,1,1,0,0] => 1

[1,0,1,0,1,1,0,0,1,0] => 2

[1,0,1,0,1,1,0,1,0,0] => 1

[1,0,1,0,1,1,1,0,0,0] => 1

[1,0,1,1,0,0,1,0,1,0] => 2

[1,0,1,1,0,0,1,1,0,0] => 2

[1,0,1,1,0,1,0,0,1,0] => 1

[1,0,1,1,0,1,0,1,0,0] => 2

[1,0,1,1,0,1,1,0,0,0] => 1

[1,0,1,1,1,0,0,0,1,0] => 2

[1,0,1,1,1,0,0,1,0,0] => 2

[1,0,1,1,1,0,1,0,0,0] => 1

[1,0,1,1,1,1,0,0,0,0] => 1

[1,1,0,0,1,0,1,0,1,0] => 1

[1,1,0,0,1,0,1,1,0,0] => 1

[1,1,0,0,1,1,0,0,1,0] => 2

[1,1,0,0,1,1,0,1,0,0] => 1

[1,1,0,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,0,1,0,1,0] => 1

[1,1,0,1,0,0,1,1,0,0] => 1

[1,1,0,1,0,1,0,0,1,0] => 2

[1,1,0,1,0,1,0,1,0,0] => 2

[1,1,0,1,0,1,1,0,0,0] => 2

[1,1,0,1,1,0,0,0,1,0] => 2

[1,1,0,1,1,0,0,1,0,0] => 1

[1,1,0,1,1,0,1,0,0,0] => 2

[1,1,0,1,1,1,0,0,0,0] => 1

[1,1,1,0,0,0,1,0,1,0] => 1

[1,1,1,0,0,0,1,1,0,0] => 1

[1,1,1,0,0,1,0,0,1,0] => 1

[1,1,1,0,0,1,0,1,0,0] => 2

[1,1,1,0,0,1,1,0,0,0] => 1

[1,1,1,0,1,0,0,0,1,0] => 1

[1,1,1,0,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,1,0,0,0] => 2

[1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,0,0,0,0,1,0] => 1

[1,1,1,1,0,0,0,1,0,0] => 1

[1,1,1,1,0,0,1,0,0,0] => 1

[1,1,1,1,0,1,0,0,0,0] => 1

[1,1,1,1,1,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => 1

[1,0,1,0,1,0,1,1,1,0,0,0] => 1

[1,0,1,0,1,1,0,0,1,0,1,0] => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => 1

[1,0,1,0,1,1,0,1,0,1,0,0] => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => 2

[1,0,1,0,1,1,1,0,0,1,0,0] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => 1

[1,0,1,1,0,1,0,1,0,0,1,0] => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => 2

[1,0,1,1,1,0,0,0,1,1,0,0] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => 2

[1,0,1,1,1,0,1,0,1,0,0,0] => 2

[1,0,1,1,1,0,1,1,0,0,0,0] => 1

>>> Load all 196 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 2\ q^{2}$

$F_{5} = 1 + 26\ q + 15\ q^{2}$

$F_{6} = 1 + 57\ q + 69\ q^{2} + 5\ q^{3}$

Description

The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.
For the projective dimension see St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. and for 1-regularity see St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra.. After applying the inverse zeta map Mp00032inverse zeta map, this statistic matches the number of rises of length at least 2 St000659The number of rises of length at least 2 of a Dyck path..

Code

def statistic(D):
    c = [ i+2 for i in reversed(D.to_area_sequence()) ] + [1]
    d = [1] + [ i+2 for i in D.reverse().to_area_sequence() ]
    return sum(1 for i in range(len(c)-1) if c[i]-c[i+1] == 1 and d[i+1]-d[i] != 1 )

#CodeLanguage: GAP

DeclareOperation("has1regularcondition",[IsList]);

InstallMethod(has1regularcondition, "for a representation of a quiver", [IsList],0,function(LIST)

local M, n, f, N, i, h,A,g,r,L,LL,subsets1,subsets2,W,simA,G1,G2,G3,g1,g2,g3,WU,O,OF,RegA;

A:=LIST[1];
M:=LIST[2];
RegA:=DirectSumOfQPAModules(IndecProjectiveModules(A));
g1:=ProjDimensionOfModule(M,30)-1;
g2:=Size(HomOverAlgebra(M,RegA));
g3:=Size(ExtOverAlgebra(M,RegA)[2])-1;
return([g1,g2,g3]);

end);

# has 1-redular condition iff output is [0,0,0].


DeclareOperation("numberofsim1ausreg",[IsList]);

InstallMethod(numberofsim1ausreg, "for a representation of a quiver", [IsList],0,function(LIST)

local M, n, f, N, i, h,A,g,r,L,LL,subsets1,subsets2,W,simA,G1,G2,G3,g1,g2,g3,WU,O,OF,RegA;

A:=LIST[1];
simA:=Filtered(SimpleModules(A),x->ProjDimensionOfModule(x,1)<=1);
WU:=Filtered(simA,x->has1regularcondition([A,x])=[0,0,0]);
return(Size(WU));

end);

DeclareOperation("allsimplesprojdim1are1reg",[IsList]);

InstallMethod(allsimplesprojdim1are1reg, "for a representation of a quiver", [IsList],0,function(LIST)

local M, n, f, N, i, h,A,g,r,L,LL,subsets1,subsets2,W,simA,G1,G2,G3,g1,g2,g3,WU,O,OF,RegA,UU,UU2;

A:=LIST[1];
UU:=numberofsim1ausreg([A]);
UU2:=Filtered(SimpleModules(A),x->ProjDimensionOfModule(x,1)=1);
return(Size(UU2)-UU);

end);

DeclareOperation("numberofsim1ausregnot",[IsList]);

InstallMethod(numberofsim1ausregnot, "for a representation of a quiver", [IsList],0,function(LIST)

local M, n, f, N, i, h,A,g,r,L,LL,subsets1,subsets2,W,simA,G1,G2,G3,g1,g2,g3,WU,O,OF,RegA,simA2;

A:=LIST[1];
simA2:=Filtered(SimpleModules(A),x->ProjDimensionOfModule(x,1)=1);
WU:=Filtered(simA2,x->has1regularcondition([A,x])=[0,0,0]);
return(Size(simA2)-Size(WU));

end);

Created

Aug 08, 2018 at 21:27 by Rene Marczinzik

Updated

Sep 05, 2018 at 10:32 by Christian Stump