Your data matches 301 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001003
(load all 8 compositions to match this statistic)
Mapping
St001003: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 3 = 1 + 2
[1,0,1,0]
 => 4 = 2 + 2
[1,1,0,0]
 => 6 = 4 + 2
[1,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,0,1,0]
 => 7 = 5 + 2
[1,1,0,1,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,0]
 => 6 = 4 + 2
Description
The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001688
(load all 9 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => 4
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 5
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
Description
The sum of the squares of the heights of the peaks of a Dyck path.
Matching statistic: St001254
(load all 16 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 6 = 4 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 7 = 5 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 6 = 4 + 2
Description
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.
Matching statistic: St000059
(load all 4 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [[1],[2]]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 4
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 3
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 5
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 5
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4
Description
The inversion number of a standard tableau as defined by Haglund and Stevens. Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
Matching statistic: St000721
(load all 5 compositions to match this statistic)
Mapping
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000721: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [(1,2)]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 4
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 5
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 5
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 4
Description
The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching. Sundaram's map sends a perfect matching on $1,\dots,2n$ to a oscillating tableau, a sequence of $n$ partitions, starting and ending with the empty partition and where two consecutive partitions differ by precisely one cell. This statistic is the sum of the sizes of these partitions, called the weight of the perfect matching in [1].
Matching statistic: St001127
(load all 3 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001127: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [1,1] => [1,1]
 => 2
[1,1,0,0]
 => [2] => [2]
 => 4
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 3
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 5
[1,1,0,1,0,0]
 => [2,1] => [2,1]
 => 5
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 4
Description
The sum of the squares of the parts of a partition.
Matching statistic: St000111
(load all 4 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000111: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => 3 = 2 + 1
[1,1,0,0]
 => [2,3,1] => [3,2,1] => 5 = 4 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 4 = 3 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => 6 = 5 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,4,2,1] => 6 = 5 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => 5 = 4 + 1
Description
The sum of the descent tops (or Genocchi descents) of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_i.$$
Matching statistic: St001074
(load all 50 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001074: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [2,3,1] => 5 = 4 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 6 = 5 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 6 = 5 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5 = 4 + 1
Description
The number of inversions of the cyclic embedding of a permutation. The cyclic embedding of a permutation $\pi$ of length $n$ is given by the permutation of length $n+1$ represented in cycle notation by $(\pi_1,\ldots,\pi_n,n+1)$. This reflects in particular the fact that the number of long cycles of length $n+1$ equals $n!$. This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length $n$ equals $n!\cdot(3n-1)/12$.
Matching statistic: St001019
(load all 23 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001019: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3 = 1 + 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 6 = 4 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 3 + 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 7 = 5 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6 = 4 + 2
Description
Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001213
(load all 6 compositions to match this statistic)
Mapping
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 6 = 4 + 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 7 = 5 + 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
The following 291 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000290The major index of a binary word. St000327The number of cover relations in a poset. St000330The (standard) major index of a standard tableau. St000393The number of strictly increasing runs in a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000224The sorting index of a permutation. St000288The number of ones in a binary word. St000305The inverse major index of a permutation. St000438The position of the last up step in a Dyck path. St000471The sum of the ascent tops of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000890The number of nonzero entries in an alternating sign matrix. St000984The number of boxes below precisely one peak. St001077The prefix exchange distance of a permutation. St001249Sum of the odd parts of a partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001759The Rajchgot index of a permutation. St000005The bounce statistic of a Dyck path. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000395The sum of the heights of the peaks of a Dyck path. St000400The path length of an ordered tree. St000756The sum of the positions of the left to right maxima of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000947The major index east count of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001782The order of rowmotion on the set of order ideals of a poset. St000495The number of inversions of distance at most 2 of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000539The number of odd inversions of a permutation. St000656The number of cuts of a poset. St000728The dimension of a set partition. St001726The number of visible inversions of a permutation. St001806The upper middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. 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. St000874The position of the last double rise in a Dyck path. St000680The Grundy value for Hackendot on posets. St000878The number of ones minus the number of zeros of a binary word. St000474Dyson's crank of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000461The rix statistic of a permutation. St000873The aix statistic of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000981The length of the longest zigzag subpath. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000088The row sums of the character table of the symmetric group. St000247The number of singleton blocks of a set partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000674The number of hills of a Dyck path. St000894The trace of an alternating sign matrix. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St000145The Dyson rank of a partition. St000509The diagonal index (content) of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000699The toughness times the least common multiple of 1,. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000545The number of parabolic double cosets with minimal element being the given permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000958The number of Bruhat factorizations of a permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001684The reduced word complexity of a permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001894The depth of a signed permutation. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000538The number of even inversions of a permutation. St000649The number of 3-excedences of a permutation. St000770The major index of an integer partition when read from bottom to top. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. 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. 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 St001285The number of primes in the column sums of the two line notation of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000420The number of Dyck paths that are weakly above a Dyck path. 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. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over 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. St001651The Frankl number of a lattice. St001658The total number of rook placements on a Ferrers board. St001874Lusztig's a-function for the symmetric group. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000293The number of inversions of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000384The maximal part of the shifted composition of an integer partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000631The number of distinct palindromic decompositions of a binary word. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000936The number of even values of the symmetric group character corresponding to the partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The 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. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001389The number of partitions of the same length below the given integer partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001488The number of corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001814The number of partitions interlacing the given partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001959The product of the heights of the peaks of a Dyck path. St000010The length of the partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000475The number of parts equal to 1 in a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000683The 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000869The sum of the hook lengths of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001164Number 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001480The number of simple summands of the module J^2/J^3. St001484The number of singletons of an integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001697The shifted natural comajor index of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001873For 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). St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000478Another weight of a partition according to Alladi.