There are

(and possibly some waiting for verification)

**284 statistics**on**Dyck paths**in the database:(and possibly some waiting for verification)

St000005Dyck paths ⟶ ℤ

The bounce statistic of a Dyck path.

St000006Dyck paths ⟶ ℤ

The dinv of a Dyck path.

St000011Dyck paths ⟶ ℤ

The number of touch points (or returns) of a Dyck path.

St000012Dyck paths ⟶ ℤ

The area of a Dyck path.

St000013Dyck paths ⟶ ℤ

The height of a Dyck path.

St000014Dyck paths ⟶ ℤ

The number of parking functions supported by a Dyck path.

St000015Dyck paths ⟶ ℤ

The number of peaks of a Dyck path.

St000024Dyck paths ⟶ ℤ

The number of double up and double down steps of a Dyck path.

St000025Dyck paths ⟶ ℤ

The number of initial rises of a Dyck path.

St000026Dyck paths ⟶ ℤ

The position of the first return of a Dyck path.

St000027Dyck paths ⟶ ℤ

The major index of a Dyck path.

St000032Dyck paths ⟶ ℤ

The number of elements smaller than the given Dyck path in the Tamari Order.

St000038Dyck paths ⟶ ℤ

The product of the heights of the descending steps of a Dyck path.

St000052Dyck paths ⟶ ℤ

The number of valleys of a Dyck path not on the x-axis.

St000053Dyck paths ⟶ ℤ

The number of valleys of the Dyck path.

St000079Dyck paths ⟶ ℤ

The number of alternating sign matrices for a given Dyck path.

St000117Dyck paths ⟶ ℤ

The number of centered tunnels of a Dyck path.

St000120Dyck paths ⟶ ℤ

The number of left tunnels of a Dyck path.

St000144Dyck paths ⟶ ℤ

The pyramid weight of the Dyck path.

St000306Dyck paths ⟶ ℤ

The bounce count of a Dyck path.

St000329Dyck paths ⟶ ℤ

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

St000331Dyck paths ⟶ ℤ

The number of upper interactions of a Dyck path.

St000335Dyck paths ⟶ ℤ

The difference of lower and upper interactions.

St000340Dyck paths ⟶ ℤ

The number of non-final maximal constant sub-paths of length greater than one.

St000369Dyck paths ⟶ ℤ

The dinv deficit of a Dyck path.

St000376Dyck paths ⟶ ℤ

The bounce deficit of a Dyck path.

St000386Dyck paths ⟶ ℤ

The number of factors DDU in a Dyck path.

St000394Dyck paths ⟶ ℤ

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

St000395Dyck paths ⟶ ℤ

The sum of the heights of the peaks of a Dyck path.

St000418Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly below a Dyck path.

St000419Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly above the Dyck path, except for the path itself.

St000420Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly above a Dyck path.

St000421Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly below a Dyck path, except for the path itself.

St000438Dyck paths ⟶ ℤ

The position of the last up step in a Dyck path.

St000439Dyck paths ⟶ ℤ

The position of the first down step of a Dyck path.

St000442Dyck paths ⟶ ℤ

The maximal area to the right of an up step of a Dyck path.

St000443Dyck paths ⟶ ℤ

The number of long tunnels of a Dyck path.

St000444Dyck paths ⟶ ℤ

The length of the maximal rise of a Dyck path.

St000445Dyck paths ⟶ ℤ

The number of rises of length 1 of a Dyck path.

St000476Dyck paths ⟶ ℤ

The sum of the semi-lengths of tunnels before a valley of a Dyck path.

St000617Dyck paths ⟶ ℤ

The number of global maxima of a Dyck path.

St000645Dyck paths ⟶ ℤ

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.

St000655Dyck paths ⟶ ℤ

The length of the minimal rise of a Dyck path.

St000658Dyck paths ⟶ ℤ

The number of rises of length 2 of a Dyck path.

St000659Dyck paths ⟶ ℤ

The number of rises of length at least 2 of a Dyck path.

St000660Dyck paths ⟶ ℤ

The number of rises of length at least 3 of a Dyck path.

St000661Dyck paths ⟶ ℤ

The number of rises of length 3 of a Dyck path.

St000674Dyck paths ⟶ ℤ

The number of hills of a Dyck path.

St000675Dyck paths ⟶ ℤ

The number of centered multitunnels of a Dyck path.

St000676Dyck paths ⟶ ℤ

The number of odd rises of a Dyck path.

St000678Dyck paths ⟶ ℤ

The number of up steps after the last double rise of a Dyck path.

St000683Dyck paths ⟶ ℤ

The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.

St000684Dyck paths ⟶ ℤ

The global dimension of the LNakayama algebra associated to a Dyck path.

St000685Dyck paths ⟶ ℤ

The dominant dimension of the LNakayama algebra associated to a Dyck path.

St000686Dyck paths ⟶ ℤ

The finitistic dominant dimension of a Dyck path.

St000687Dyck paths ⟶ ℤ

The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.

St000688Dyck paths ⟶ ℤ

The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path.

St000689Dyck paths ⟶ ℤ

The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.

St000790Dyck paths ⟶ ℤ

The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.

St000791Dyck paths ⟶ ℤ

The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.

St000874Dyck paths ⟶ ℤ

The position of the last double rise in a Dyck path.

St000920Dyck paths ⟶ ℤ

The logarithmic height of a Dyck path.

St000930Dyck paths ⟶ ℤ

The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.

St000931Dyck paths ⟶ ℤ

The number of occurrences of the pattern UUU in a Dyck path.

St000932Dyck paths ⟶ ℤ

The number of occurrences of the pattern UDU in a Dyck path.

St000946Dyck paths ⟶ ℤ

The sum of the skew hook positions in a Dyck path.

St000947Dyck paths ⟶ ℤ

The major index east count of a Dyck path.

St000949Dyck paths ⟶ ℤ

Gives the number of generalised tilting modules of the corresponding LNakayama algebra.

St000950Dyck paths ⟶ ℤ

Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.

St000951Dyck paths ⟶ ℤ

The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra.

St000952Dyck paths ⟶ ℤ

Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers.

St000953Dyck paths ⟶ ℤ

The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers.

St000954Dyck paths ⟶ ℤ

Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.

St000955Dyck paths ⟶ ℤ

Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.

St000964Dyck paths ⟶ ℤ

Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra.

St000965Dyck paths ⟶ ℤ

The sum of the dimension of Ext^i(D(A),A) for i=1,.

St000966Dyck paths ⟶ ℤ

Number of peaks minus the global dimension of the corresponding LNakayama algebra.

St000967Dyck paths ⟶ ℤ

The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra.

St000968Dyck paths ⟶ ℤ

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$.

St000969Dyck paths ⟶ ℤ

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$.

St000970Dyck paths ⟶ ℤ

Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.

St000976Dyck paths ⟶ ℤ

The sum of the positions of double up-steps of a Dyck path.

St000977Dyck paths ⟶ ℤ

MacMahon's equal index of a Dyck path.

St000978Dyck paths ⟶ ℤ

The sum of the positions of double down-steps of a Dyck path.

St000979Dyck paths ⟶ ℤ

Half of MacMahon's equal index of a Dyck path.

St000980Dyck paths ⟶ ℤ

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.

St000981Dyck paths ⟶ ℤ

The length of the longest zigzag subpath.

St000984Dyck paths ⟶ ℤ

The number of boxes below precisely one peak.

St000998Dyck paths ⟶ ℤ

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

St000999Dyck paths ⟶ ℤ

Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

St001000Dyck paths ⟶ ℤ

Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

St001001Dyck paths ⟶ ℤ

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001002Dyck paths ⟶ ℤ

Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

St001003Dyck paths ⟶ ℤ

The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

St001006Dyck paths ⟶ ℤ

Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001007Dyck paths ⟶ ℤ

Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

St001008Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

St001009Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001010Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001011Dyck paths ⟶ ℤ

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

St001012Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.

St001013Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

St001014Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path.

St001015Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path.

St001016Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

St001017Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.

St001018Dyck paths ⟶ ℤ

Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

St001019Dyck paths ⟶ ℤ

Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

St001020Dyck paths ⟶ ℤ

Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

St001021Dyck paths ⟶ ℤ

Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.

St001022Dyck paths ⟶ ℤ

Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.

St001023Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path.

St001024Dyck paths ⟶ ℤ

Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

St001025Dyck paths ⟶ ℤ

Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path.

St001026Dyck paths ⟶ ℤ

The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.

St001027Dyck paths ⟶ ℤ

Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path.

St001028Dyck paths ⟶ ℤ

Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

St001031Dyck paths ⟶ ℤ

The height of the bicoloured Motzkin path associated with the Dyck path.

St001032Dyck paths ⟶ ℤ

The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path.

St001033Dyck paths ⟶ ℤ

The normalized area of the parallelogram polyomino associated with the Dyck path.

St001034Dyck paths ⟶ ℤ

The area of the parallelogram polyomino associated with the Dyck path.

St001035Dyck paths ⟶ ℤ

The convexity degree of the parallelogram polyomino associated with the Dyck path.

St001036Dyck paths ⟶ ℤ

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

St001037Dyck paths ⟶ ℤ

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

St001038Dyck paths ⟶ ℤ

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

St001039Dyck paths ⟶ ℤ

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

St001063Dyck paths ⟶ ℤ

Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.

St001064Dyck paths ⟶ ℤ

Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.

St001065Dyck paths ⟶ ℤ

Number of indecomposable reflexive modules in the corresponding Nakayama algebra.

St001066Dyck paths ⟶ ℤ

The number of simple reflexive modules in the corresponding Nakayama algebra.

St001067Dyck paths ⟶ ℤ

The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.

St001068Dyck paths ⟶ ℤ

Number of torsionless simple modules in the corresponding Nakayama algebra.

St001088Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

St001089Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

St001104Dyck paths ⟶ ℤ

The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group.

St001107Dyck paths ⟶ ℤ

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.

St001113Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.

St001125Dyck paths ⟶ ℤ

The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.

St001126Dyck paths ⟶ ℤ

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

St001135Dyck paths ⟶ ℤ

The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.

St001137Dyck paths ⟶ ℤ

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

St001138Dyck paths ⟶ ℤ

The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra.

St001139Dyck paths ⟶ ℤ

The number of occurrences of hills of size 2 in a Dyck path.

St001140Dyck paths ⟶ ℤ

Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra.

St001141Dyck paths ⟶ ℤ

The number of occurrences of hills of size 3 in a Dyck path.

St001142Dyck paths ⟶ ℤ

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

St001159Dyck paths ⟶ ℤ

Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.

St001161Dyck paths ⟶ ℤ

The major index north count of a Dyck path.

St001163Dyck paths ⟶ ℤ

The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra.

St001164Dyck paths ⟶ ℤ

Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules.

St001165Dyck paths ⟶ ℤ

Number of simple modules with even projective dimension in the corresponding Nakayama algebra.

St001166Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra.

St001167Dyck paths ⟶ ℤ

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.

St001169Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

St001170Dyck paths ⟶ ℤ

Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra.

St001172Dyck paths ⟶ ℤ

The number of 1-rises at odd height of a Dyck path.

St001179Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.

St001180Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension at most 1.

St001181Dyck paths ⟶ ℤ

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.

St001182Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.

St001183Dyck paths ⟶ ℤ

The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.

St001184Dyck paths ⟶ ℤ

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

St001185Dyck paths ⟶ ℤ

The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

St001186Dyck paths ⟶ ℤ

Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.

St001187Dyck paths ⟶ ℤ

The number of simple modules with grade at least one in the corresponding Nakayama algebra.

St001188Dyck paths ⟶ ℤ

The 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.

St001189Dyck paths ⟶ ℤ

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

St001190Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.

St001191Dyck paths ⟶ ℤ

Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$.

St001192Dyck paths ⟶ ℤ

The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.

St001193Dyck paths ⟶ ℤ

The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module.

St001194Dyck paths ⟶ ℤ

The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module

St001195Dyck paths ⟶ ℤ

The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.

St001196Dyck paths ⟶ ℤ

The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

St001197Dyck paths ⟶ ℤ

The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001198Dyck paths ⟶ ℤ

The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001199Dyck paths ⟶ ℤ

The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001200Dyck paths ⟶ ℤ

The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001201Dyck paths ⟶ ℤ

The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.

St001202Dyck paths ⟶ ℤ

Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.

St001203Dyck paths ⟶ ℤ

We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:

St001204Dyck paths ⟶ ℤ

Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.

St001205Dyck paths ⟶ ℤ

The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001206Dyck paths ⟶ ℤ

The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

St001210Dyck paths ⟶ ℤ

Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.

St001211Dyck paths ⟶ ℤ

The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.

St001212Dyck paths ⟶ ℤ

The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.

St001213Dyck paths ⟶ ℤ

The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.

St001215Dyck paths ⟶ ℤ

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.

St001216Dyck paths ⟶ ℤ

The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.

St001217Dyck paths ⟶ ℤ

The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.

St001218Dyck paths ⟶ ℤ

Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1.

St001219Dyck paths ⟶ ℤ

Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive.

St001221Dyck paths ⟶ ℤ

The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.

St001222Dyck paths ⟶ ℤ

Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.

St001223Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.

St001224Dyck paths ⟶ ℤ

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.

St001225Dyck paths ⟶ ℤ

The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.

St001226Dyck paths ⟶ ℤ

The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.

St001227Dyck paths ⟶ ℤ

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

St001228Dyck paths ⟶ ℤ

The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.

St001229Dyck paths ⟶ ℤ

The vector space dimension of the first extension group between the Jacobson radical J and J^2.

St001230Dyck paths ⟶ ℤ

The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.

St001231Dyck paths ⟶ ℤ

The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.

St001232Dyck paths ⟶ ℤ

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

St001233Dyck paths ⟶ ℤ

The number of indecomposable 2-dimensional modules with projective dimension one.

St001234Dyck paths ⟶ ℤ

The number of indecomposable three dimensional modules with projective dimension one.

St001237Dyck paths ⟶ ℤ

The number of simple modules with injective dimension at most one or dominant dimension at least one.

St001238Dyck paths ⟶ ℤ

The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S.

St001239Dyck paths ⟶ ℤ

The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.

St001240Dyck paths ⟶ ℤ

The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra

St001241Dyck paths ⟶ ℤ

The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one.

St001242Dyck paths ⟶ ℤ

The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path.

St001243Dyck paths ⟶ ℤ

The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path.

St001244Dyck paths ⟶ ℤ

The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.

St001253Dyck paths ⟶ ℤ

The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.

St001254Dyck paths ⟶ ℤ

The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.

St001255Dyck paths ⟶ ℤ

The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

St001256Dyck paths ⟶ ℤ

Number of simple reflexive modules that are 2-stable reflexive.

St001257Dyck paths ⟶ ℤ

The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

St001258Dyck paths ⟶ ℤ

Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.

St001259Dyck paths ⟶ ℤ

The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.

St001264Dyck paths ⟶ ℤ

The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.

St001265Dyck paths ⟶ ℤ

The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.

St001266Dyck paths ⟶ ℤ

The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra.

St001273Dyck paths ⟶ ℤ

The projective dimension of the first term in an injective coresolution of the regular module.

St001274Dyck paths ⟶ ℤ

The number of indecomposable injective modules with projective dimension equal to two.

St001275Dyck paths ⟶ ℤ

The projective dimension of the second term in a minimal injective coresolution of the regular module.

St001276Dyck paths ⟶ ℤ

The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.

St001278Dyck paths ⟶ ℤ

The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.

St001289Dyck paths ⟶ ℤ

The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero.

St001290Dyck paths ⟶ ℤ

The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.

St001291Dyck paths ⟶ ℤ

The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.

St001292Dyck paths ⟶ ℤ

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.

St001294Dyck paths ⟶ ℤ

The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.

St001295Dyck paths ⟶ ℤ

Gives the vector space dimension of the homomorphism space between J^2 and J^2.

St001296Dyck paths ⟶ ℤ

The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra.

St001297Dyck paths ⟶ ℤ

The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.

St001299Dyck paths ⟶ ℤ

The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.

St001314Dyck paths ⟶ ℤ

The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.

St001348Dyck paths ⟶ ℤ

The bounce of the parallelogram polyomino associated with the Dyck path.

St001361Dyck paths ⟶ ℤ

The number of lattice paths of the same length that stay weakly above a Dyck path.

St001418Dyck paths ⟶ ℤ

Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.

St001431Dyck paths ⟶ ℤ

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.

St001471Dyck paths ⟶ ℤ

The magnitude of a Dyck path.

St001473Dyck paths ⟶ ℤ

The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra.

St001480Dyck paths ⟶ ℤ

The number of simple summands of the module J^2/J^3.

St001481Dyck paths ⟶ ℤ

The minimal height of a peak of a Dyck path.

St001483Dyck paths ⟶ ℤ

The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.

St001492Dyck paths ⟶ ℤ

The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.

St001493Dyck paths ⟶ ℤ

The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.

St001498Dyck paths ⟶ ℤ

The normalised height of a Nakayama algebra with magnitude 1.

St001499Dyck paths ⟶ ℤ

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.

St001500Dyck paths ⟶ ℤ

The global dimension of magnitude 1 Nakayama algebras.

St001501Dyck paths ⟶ ℤ

The dominant dimension of magnitude 1 Nakayama algebras.

St001502Dyck paths ⟶ ℤ

The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras.

St001503Dyck paths ⟶ ℤ

The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.

St001504Dyck paths ⟶ ℤ

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

St001505Dyck paths ⟶ ℤ

The number of elements generated by the Dyck path as a map in the full transformation monoid.

St001506Dyck paths ⟶ ℤ

Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.

St001507Dyck paths ⟶ ℤ

The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.

St001508Dyck paths ⟶ ℤ

The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.

St001509Dyck paths ⟶ ℤ

The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary.

St001514Dyck paths ⟶ ℤ

The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.

St001515Dyck paths ⟶ ℤ

The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).

St001523Dyck paths ⟶ ℤ

The degree of symmetry of a Dyck path.

St001526Dyck paths ⟶ ℤ

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

St001530Dyck paths ⟶ ℤ

The depth of a Dyck path.

St001531Dyck paths ⟶ ℤ

Number of partial orders contained in the poset determined by the Dyck path.

St001553Dyck paths ⟶ ℤ

The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.

St001584Dyck paths ⟶ ℤ

The area statistic between a Dyck path and its bounce path.

St001594Dyck paths ⟶ ℤ

The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied.

St001643Dyck paths ⟶ ℤ

The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.

St001650Dyck paths ⟶ ℤ

The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path.

St001669Dyck paths ⟶ ℤ

The number of single rises in a Dyck path.

St001688Dyck paths ⟶ ℤ

The sum of the squares of the heights of the peaks of a Dyck path.

St001732Dyck paths ⟶ ℤ

The number of peaks visible from the left.

St001733Dyck paths ⟶ ℤ

The number of weak left to right maxima of a Dyck path.

St001786Dyck paths ⟶ ℤ

The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.

St001800Dyck paths ⟶ ℤ

The number of 3-Catalan paths having this Dyck path as first and last coordinate projections.

St001808Dyck paths ⟶ ℤ

The box weight or horizontal decoration of a Dyck path.

St001809Dyck paths ⟶ ℤ

The index of the step at the first peak of maximal height in a Dyck path.

St001872Dyck paths ⟶ ℤ

The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra.

St001873Dyck paths ⟶ ℤ

For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules).