There are 292 statistics
on Dyck paths
in the database:
(and possibly some waiting for verification)
(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 linear Nakayama algebra corresponding to a Dyck path.
St000686Dyck paths ⟶ ℤ
The finitistic dominant dimension of a Dyck path.
St000687Dyck paths ⟶ ℤ
The dimension of $\operatorname{Hom}(I,P)$ for the linear Nakayama algebra corresponding to a Dyck path.
St000688Dyck paths ⟶ ℤ
The global dimension minus the dominant dimension of the linear Nakayama algebra corresponding to a Dyck path.
St000689Dyck paths ⟶ ℤ
The maximal $n$ such that the minimal generator-cogenerator module in the linear Nakayama algebra corresponding to 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 linear Nakayama algebra corresponding to a Dyck path.
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
The number of generalised tilting modules of the linear Nakayama algebra corresponding to a Dyck path.
St000950Dyck paths ⟶ ℤ
The number of tilting modules of the linear Nakayama algebra corresponding to a Dyck path.
St000951Dyck paths ⟶ ℤ
The dimension of $\operatorname{Ext}^{1}(D(A),A)$ for the linear Nakayama algebra corresponding to a Dyck path.
St000952Dyck paths ⟶ ℤ
The number of irreducible factors over $\mathbb{Q}$ of the Coxeter polynomial of the linear Nakayama algebra corresponding to a Dyck path.
St000953Dyck paths ⟶ ℤ
The largest degree of an irreducible factor over $\mathbb{Q}$ of the Coxeter polynomial of the linear Nakayama algebra corresponding to a Dyck path.
St000954Dyck paths ⟶ ℤ
The number of integers $i > 0$ such that $\operatorname{Ext}^i(D(A),A)=0$ for the linear Nakayama algebra corresponding to a Dyck path.
St000955Dyck paths ⟶ ℤ
The number of integers $i > 0$ such that $\operatorname{Ext}^i(D(A),A) > 0$ for the linear Nakayama algebra corresponding to a Dyck path.
St000964Dyck paths ⟶ ℤ
The dimension of $\operatorname{Ext}^g(D(A),A)$ for the linear Nakayama algebra corresponding to a Dyck path, where $g$ is the global dimension of that algebra.
St000965Dyck paths ⟶ ℤ
The sum of the dimensions of $\operatorname{Ext}^i(D(A),A)$ for $i=1,\ldots,g$, where $g$ is the global dimension of the linear Nakayama algebra corresponding to a Dyck path.
St000966Dyck paths ⟶ ℤ
The number of peaks of a Dyck path minus the global dimension of the corresponding linear Nakayama algebra.
St000967Dyck paths ⟶ ℤ
The value $p(1)$ for the Coxeter polynomial $p$ of the linear Nakayama algebra corresponding to a Dyck path.
St000968Dyck paths ⟶ ℤ
The dominant dimension of the cyclic Nakayama algebra obtained from the linear Nakayama algebra corresponding to a Dyck path.
St000969Dyck paths ⟶ ℤ
The global dimension of the cyclic Nakayama algebra obtained from the linear Nakayama algebra corresponding to a Dyck path.
St000970Dyck paths ⟶ ℤ
The number of peaks of a Dyck path minus the dominant dimension of the corresponding linear Nakayama 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
The number of indecomposable projective modules with injective dimension less than or equal to the dominant dimension in the linear Nakayama algebra corresponding to a Dyck path.
St000999Dyck paths ⟶ ℤ
The number of indecomposable projective modules with injective dimension equal to the global dimension in the linear Nakayama algebra corresponding to a Dyck path.
St001000Dyck paths ⟶ ℤ
The number of indecomposable modules with projective dimension equal to the global dimension in the linear Nakayama algebra corresponding to a Dyck path.
St001001Dyck paths ⟶ ℤ
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the linear Nakayama algebra corresponding to a Dyck path.
St001002Dyck paths ⟶ ℤ
The number of indecomposable modules with projective and injective dimension at most one in the linear Nakayama algebra corresponding to a Dyck path.
St001003Dyck paths ⟶ ℤ
The number of indecomposable modules with projective dimension at most one in the linear Nakayama algebra corresponding to a Dyck path.
St001006Dyck paths ⟶ ℤ
The number of simple modules with projective dimension equal to the global dimension of the linear Nakayama algebra corresponding to a Dyck path.
St001007Dyck paths ⟶ ℤ
The number of simple modules with projective dimension one in the linear Nakayama algebra corresponding to a Dyck path.
St001008Dyck paths ⟶ ℤ
The number of indecomposable injective modules with projective dimension one in the linear Nakayama algebra corresponding to a Dyck path.
St001009Dyck paths ⟶ ℤ
The number of indecomposable injective modules with projective dimension equal to the global dimension of the linear Nakayama algebra corresponding to a Dyck path.
St001010Dyck paths ⟶ ℤ
The number of indecomposable injective modules with projective dimension equal to the global dimension minus one of the linear Nakayama algebra corresponding to a Dyck path.
St001011Dyck paths ⟶ ℤ
The number of simple modules with projective dimension two in the linear Nakayama algebra corresponding to a 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
The number of indecomposable injective modules with codominant dimension equal to the global dimension in the linear Nakayama algebra corresponding to a Dyck path.
St001014Dyck paths ⟶ ℤ
The number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the linear Nakayama algebra corresponding to a 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
The number of indecomposable injective modules with codominant dimension at most one in the linear Nakayama algebra corresponding to a 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
The sum of the projective dimensions of the indecomposable injective modules of the linear Nakayama algebra corresponding to a Dyck path.
St001019Dyck paths ⟶ ℤ
The sum of the projective dimensions of the simple modules of the linear Nakayama algebra corresponding to a Dyck path.
St001020Dyck paths ⟶ ℤ
The sum of the codominant dimensions of the non-projective indecomposable injective modules of the linear Nakayama algebra corresponding to a Dyck path.
St001021Dyck paths ⟶ ℤ
The sum of the differences between the projective and codominant dimensions of the non-projective indecomposable injective modules of the linear Nakayama algebra corresponding to a Dyck path.
St001022Dyck paths ⟶ ℤ
The number of simple modules with projective dimension three in the linear Nakayama algebra corresponding to a Dyck path.
St001023Dyck paths ⟶ ℤ
The number of simple modules with projective dimension at most three in the linear Nakayama algebra corresponding to a Dyck path.
St001024Dyck paths ⟶ ℤ
The maximum of the dominant dimensions of the simple modules of the linear Nakayama algebra corresponding to a Dyck path.
St001025Dyck paths ⟶ ℤ
The number of simple modules with projective dimension four in the linear Nakayama algebra corresponding to a Dyck path.
St001026Dyck paths ⟶ ℤ
The difference between the maximum and the minimum of the projective dimensions of the indecomposable non-projective injective modules of the linear Nakayama algebra corresponding to a Dyck path.
St001027Dyck paths ⟶ ℤ
The number of simple modules with projective dimension equal to injective dimension in the linear Nakayama algebra corresponding to a Dyck path.
St001028Dyck paths ⟶ ℤ
The number of simple modules with injective dimension equal to the dominant dimension in the linear Nakayama algebra corresponding to a 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
The number of 3-torsionfree simple modules in the linear Nakayama algebra corresponding to a Dyck path.
St001064Dyck paths ⟶ ℤ
The number of simple modules of the linear Nakayama algebra corresponding to a Dyck path 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 of the linear Nakayama algebra corresponding to a Dyck path.
St001067Dyck paths ⟶ ℤ
The number of simple modules of dominant dimension at least two in the linear Nakayama algebra corresponding to a Dyck path.
St001068Dyck paths ⟶ ℤ
The number of torsionless simple modules in the linear Nakayama algebra corresponding to a Dyck path.
St001088Dyck paths ⟶ ℤ
The number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the linear Nakayama algebra corresponding to a Dyck path.
St001089Dyck paths ⟶ ℤ
The number of indecomposable projective non-injective modules of the linear Nakayama algebra corresponding to a Dyck path, minus the number of indecomposable projective non-injective modules whose dominant dimension equals their injective dimension.
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
The number of indecomposable projective non-injective modules of the linear Nakayama algebra corresponding to a Dyck path that have reflexive Auslander–Reiten sequences.
St001125Dyck paths ⟶ ℤ
The number of 2-regular simple modules of the linear Nakayama algebra corresponding to a Dyck path.
St001126Dyck paths ⟶ ℤ
The number of 1-regular simple modules of the linear Nakayama algebra corresponding to a Dyck path.
St001135Dyck paths ⟶ ℤ
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
St001137Dyck paths ⟶ ℤ
The number of 3-regular simple modules of the linear Nakayama algebra corresponding to a Dyck path.
St001138Dyck paths ⟶ ℤ
The number of indecomposable modules of the linear Nakayama algebra corresponding to a Dyck path whose projective dimension or injective dimension is at most one.
St001139Dyck paths ⟶ ℤ
The number of occurrences of hills of size 2 in a Dyck path.
St001140Dyck paths ⟶ ℤ
The number of indecomposable modules of the linear Nakayama algebra corresponding to a Dyck path whose projective dimension and injective dimension are at least two.
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
The number of simple modules of the linear Nakayama algebra corresponding to a Dyck path whose dominant dimension equals the global dimension of the algebra.
St001161Dyck paths ⟶ ℤ
The major index north count of a Dyck path.
St001163Dyck paths ⟶ ℤ
The number of simple modules of dominant dimension at least three in the linear Nakayama algebra corresponding to a Dyck path.
St001164Dyck paths ⟶ ℤ
The number of indecomposable injective modules whose socle has projective dimension at most the global dimension of the algebra minus one of the linear Nakayama algebra corresponding to a Dyck path, minus the number of indecomposable projective-injective modules.
St001165Dyck paths ⟶ ℤ
The number of simple modules with even projective dimension of the linear Nakayama algebra corresponding to a Dyck path.
St001166Dyck paths ⟶ ℤ
The number of indecomposable projective non-injective modules whose dominant dimension equals the global dimension of the linear Nakayama algebra corresponding to a Dyck path, plus the number of indecomposable projective-injective modules.
St001167Dyck paths ⟶ ℤ
The number of simple modules that occur as the top of an indecomposable non-projective reflexive module in the linear Nakayama algebra corresponding to a Dyck path.
St001169Dyck paths ⟶ ℤ
The number of simple modules with projective dimension at least two of the linear Nakayama algebra corresponding to a Dyck path.
St001170Dyck paths ⟶ ℤ
The number of indecomposable injective modules whose socle has projective dimension at most the global dimension of the algebra minus one of the linear Nakayama algebra corresponding to a Dyck path.
St001172Dyck paths ⟶ ℤ
The number of 1-rises at odd height of a Dyck path.
St001179Dyck paths ⟶ ℤ
The number of indecomposable injective modules with projective dimension at most two of the linear Nakayama algebra corresponding to a Dyck path.
St001180Dyck paths ⟶ ℤ
The number of indecomposable injective modules with projective dimension at most one of the linear Nakayama algebra corresponding to a Dyck path.
St001181Dyck paths ⟶ ℤ
The number of indecomposable injective modules of grade at least three in the linear Nakayama algebra corresponding to a Dyck path.
St001182Dyck paths ⟶ ℤ
The number of indecomposable injective modules with codominant dimension at least two of the linear Nakayama algebra corresponding to a Dyck path.
St001183Dyck paths ⟶ ℤ
The maximum of $\operatorname{projdim}(S)+\operatorname{injdim}(S)$ over all simple modules $S$ of the linear Nakayama algebra corresponding to a Dyck path.
St001184Dyck paths ⟶ ℤ
The number of indecomposable injective modules of grade at least one in the linear Nakayama algebra corresponding to a Dyck path.
St001185Dyck paths ⟶ ℤ
The number of indecomposable injective modules of grade at least two in the linear Nakayama algebra corresponding to a Dyck path.
St001186Dyck paths ⟶ ℤ
The number of simple modules of grade at least three in the linear Nakayama algebra corresponding to a Dyck path.
St001187Dyck paths ⟶ ℤ
The number of simple modules of grade at least one in the linear Nakayama algebra corresponding to a Dyck path.
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 smallest natural number $n$ such that $D(A)^{\otimes n} = 0$ for the linear Nakayama algebra $A$ corresponding to a Dyck path.
St001291Dyck paths ⟶ ℤ
The number of indecomposable summands of $D(A)\otimes D(A)$ of the linear Nakayama algebra $A$ corresponding to a Dyck path.
St001292Dyck paths ⟶ ℤ
The injective dimension of $D(A)\otimes_A D(A)$ for the linear Nakayama algebra $A$ corresponding to a Dyck path.
St001294Dyck paths ⟶ ℤ
The maximal torsionfree index of a simple non-projective module of the linear Nakayama algebra corresponding to a Dyck path.
St001295Dyck paths ⟶ ℤ
The vector space dimension of $\operatorname{Hom}_A(J^2,J^2)$ for the linear Nakayama algebra $A$ corresponding to a Dyck path.
St001296Dyck paths ⟶ ℤ
The maximal torsionfree index of an indecomposable non-projective module in the linear Nakayama algebra corresponding to a Dyck path.
St001297Dyck paths ⟶ ℤ
The number of indecomposable non-injective projective modules of the linear Nakayama algebra corresponding to a Dyck path, minus the number of such modules that have reflexive Auslander–Reiten sequences.
St001299Dyck paths ⟶ ℤ
The product of all nonzero projective dimensions of simple modules of the linear Nakayama algebra corresponding to a Dyck path.
St001314Dyck paths ⟶ ℤ
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the linear Nakayama algebra corresponding to a Dyck path.
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 $J^2/J^3$ in the linear Nakayama algebra corresponding to a Dyck path.
St001481Dyck paths ⟶ ℤ
The minimal height of a peak of a Dyck path.
St001483Dyck paths ⟶ ℤ
The number of simple modules $S$ that occur in the socle of the regular module $A$ of the linear Nakayama algebra corresponding to a Dyck path and satisfy $\operatorname{Ext}^1_A(S,A)=0$.
St001492Dyck paths ⟶ ℤ
The number of simple modules of the linear Nakayama algebra corresponding to a Dyck path that do not appear in the socle of the regular module, or have no nontrivial self-extensions with the regular module.
St001493Dyck paths ⟶ ℤ
The number of simple modules whose projective dimension is even and is maximal among the even projective dimensions occurring for simple modules of the linear Nakayama algebra corresponding to a Dyck path.
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
Half the sum of the projective dimensions of the simple modules of the linear Nakayama algebra corresponding to a Dyck path that have even projective dimension.
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 a 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 of even projective dimension in the linear Nakayama algebra corresponding to a Dyck path.
St001873Dyck paths ⟶ ℤ
Half the rank of the matrix $C^T-C$, where $C$ is the Coxeter matrix of the linear Nakayama algebra corresponding to a Dyck path.
St001910Dyck paths ⟶ ℤ
The height of the middle non-run of a Dyck path.
St001929Dyck paths ⟶ ℤ
The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path.
St001932Dyck paths ⟶ ℤ
The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition.
St001955Dyck paths ⟶ ℤ
The number of natural descents for set-valued two row standard Young tableaux.
St001956Dyck paths ⟶ ℤ
The comajor index for set-valued two-row standard Young tableaux.
St001959Dyck paths ⟶ ℤ
The product of the heights of the peaks of a Dyck path.
St001966Dyck paths ⟶ ℤ
Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path).
St001983Dyck paths ⟶ ℤ
The number of indecomposable injective modules that are pure in the linear Nakayama algebra corresponding to a Dyck path.