Your data matches 178 different statistics following compositions of up to 3 maps.
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Matching statistic: St001247
(load all 2 compositions to match this statistic)
Mapping
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 2 = 3 - 1
[3]
 => 1 = 2 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 3 = 4 - 1
[4]
 => 1 = 2 - 1
[3,1]
 => 2 = 3 - 1
[2,2]
 => 0 = 1 - 1
[2,1,1]
 => 2 = 3 - 1
[1,1,1,1]
 => 4 = 5 - 1
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001816
(load all 3 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => 1 = 2 - 1
[2]
 => [[1,2]]
 => 2 = 3 - 1
[1,1]
 => [[1],[2]]
 => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => 3 = 4 - 1
[2,1]
 => [[1,2],[3]]
 => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => 1 = 2 - 1
[4]
 => [[1,2,3,4]]
 => 4 = 5 - 1
[3,1]
 => [[1,2,3],[4]]
 => 2 = 3 - 1
[2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0 = 1 - 1
Description
Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St000145
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 2 - 2
[2]
 => [1,1]
 => -1 = 1 - 2
[1,1]
 => [2]
 => 1 = 3 - 2
[3]
 => [2,1]
 => 0 = 2 - 2
[2,1]
 => [2,1]
 => 0 = 2 - 2
[1,1,1]
 => [3]
 => 2 = 4 - 2
[4]
 => [3,1]
 => 1 = 3 - 2
[3,1]
 => [2,2]
 => 0 = 2 - 2
[2,2]
 => [2,1,1]
 => -1 = 1 - 2
[2,1,1]
 => [3,1]
 => 1 = 3 - 2
[1,1,1,1]
 => [4]
 => 3 = 5 - 2
Description
The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000885
(load all 19 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 01 => 2
[2]
 => 100 => 101 => 1
[1,1]
 => 110 => 011 => 3
[3]
 => 1000 => 0101 => 2
[2,1]
 => 1010 => 1001 => 2
[1,1,1]
 => 1110 => 0111 => 4
[4]
 => 10000 => 10101 => 1
[3,1]
 => 10010 => 01101 => 3
[2,2]
 => 1100 => 1011 => 2
[2,1,1]
 => 10110 => 10001 => 3
[1,1,1,1]
 => 11110 => 01111 => 5
Description
The number of critical steps in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation. The distribution of this statistic on words of length $n$ is $$ (n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}. $$
Matching statistic: St001210
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
Description
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.
Matching statistic: St000288
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Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 10 => 1 = 2 - 1
[1,1,1]
 => 111 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 01 => 1 = 2 - 1
[2,1,1]
 => 011 => 110 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 1111 => 4 = 5 - 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 10 => 1 = 2 - 1
[1,1,1]
 => 111 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 01 => 1 = 2 - 1
[2,1,1]
 => 011 => 110 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 1111 => 4 = 5 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000445
(load all 7 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => 1 = 2 - 1
[2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000475
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => 1 = 2 - 1
[2]
 => [1,1]
 => [2]
 => 0 = 1 - 1
[1,1]
 => [2]
 => [1,1]
 => 2 = 3 - 1
[3]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[2,1]
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,1]
 => [3]
 => [1,1,1]
 => 3 = 4 - 1
[4]
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[3,1]
 => [2,2]
 => [2,2]
 => 0 = 1 - 1
[2,2]
 => [2,1,1]
 => [3,1]
 => 1 = 2 - 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 4 = 5 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001017
(load all 9 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => 1 = 2 - 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 3 - 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
Description
Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.
The following 168 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000025The number of initial rises of a Dyck path. St000044The number of vertices of the unicellular map given by a perfect matching. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000676The number of odd rises of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000702The number of weak deficiencies of a permutation. St000808The number of up steps of the associated bargraph. St000991The number of right-to-left minima of a permutation. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001202Call 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. St001497The position of the largest weak excedence of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000022The number of fixed points of a permutation. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000148The number of odd parts of a partition. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000241The number of cyclical small excedances. St000295The length of the border of a binary word. St000309The number of vertices with even degree. St000338The number of pixed points of a permutation. St000389The number of runs of ones of odd length in a binary word. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000691The number of changes of a binary word. St000819The propagating number of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We 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 $c_{n-1}$. St000992The alternating sum of the parts of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001223Number 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001498The normalised height of a Nakayama algebra with magnitude 1. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001948The number of augmented double ascents of a permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St000878The number of ones minus the number of zeros of a binary word. St000674The number of hills of a Dyck path. St000248The number of anti-singletons of a set partition. St000461The rix statistic of a permutation. St000873The aix statistic of a permutation. St000894The trace of an alternating sign matrix. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000806The semiperimeter of the associated bargraph. St000884The number of isolated descents of a permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000937The number of positive values of the symmetric group character corresponding to the partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000045The number of linear extensions of a binary tree. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001346The number of parking functions that give the same permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000090The variation of a composition. St000259The diameter of a connected graph. St000014The number of parking functions supported by a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000294The number of distinct factors of a binary word. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000759The smallest missing part in an integer partition. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We 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 $c_{n−1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001166Number 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001198The 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 $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001206The 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001226The 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001492The 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. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001523The degree of symmetry of a Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001814The number of partitions interlacing the given partition. St001885The number of binary words with the same proper border set. St000454The largest eigenvalue of a graph if it is integral. St000850The number of 1/2-balanced pairs in a poset. St000455The second largest eigenvalue of a graph if it is integral.