Your data matches 322 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000032
(load all 2 compositions to match this statistic)
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000032: 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]
 => 2 = 3 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 9 = 10 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 6 = 7 - 1
[]
 => []
 => [1,0]
 => 1 = 2 - 1
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000231
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000231: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 2 = 3 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 5 = 6 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3 = 4 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 9 = 10 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 6 = 7 - 1
[]
 => [1,0]
 => {{1}}
 => 1 = 2 - 1
Description
Sum of the maximal elements of the blocks of a set partition.
Matching statistic: St001228
(load all 6 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001228: 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]
 => 2 = 3 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 5 = 6 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 9 = 10 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 7 - 1
[]
 => [1,0]
 => [1,0]
 => 1 = 2 - 1
Description
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.
Matching statistic: St001688
(load all 8 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001688: 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]
 => 2 = 3 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 5 = 6 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 9 = 10 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 7 - 1
[]
 => [1,0]
 => [1,0]
 => 1 = 2 - 1
Description
The sum of the squares of the heights of the peaks of a Dyck path.
Matching statistic: St000070
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000070: Posets ⟶ 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]
 => ([(0,1)],2)
 => 3
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 4
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[]
 => [1,0]
 => [1,0]
 => ([],1)
 => 2
Description
The number of antichains in a poset. An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable. An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St000104
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000104: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 3
[1,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[1,1,0,0]
 => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 10
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[]
 => [1] => [1] => ([],1)
 => 2
Description
The number of facets in the order polytope of this poset.
Matching statistic: St000110
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 6
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 4
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 10
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => 7
[]
 => [1,0]
 => [1,1,0,0]
 => [2,1] => 2
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000151
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000151: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 3
[1,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[1,1,0,0]
 => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 10
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[]
 => [1] => [1] => ([],1)
 => 2
Description
The number of facets in the chain polytope of the poset.
Matching statistic: St000420
(load all 4 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 3
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 6
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 10
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 7
[]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000825
(load all 3 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000825: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [2,3,1] => 3
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 6
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 4
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 10
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 7
[]
 => [1,0]
 => [1,0]
 => [2,1] => 2
Description
The sum of the major and the inverse major index of a permutation. This statistic is the sum of [[St000004]] and [[St000305]].
The following 312 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000012The area of a Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000400The path length of an ordered tree. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000520The number of patterns in a permutation. St000529The number of permutations whose descent word is the given binary word. St001161The major index north count of a Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001254The 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. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000014The number of parking functions supported by a Dyck path. St000639The number of relations in a poset. St000756The sum of the positions of the left to right maxima of a permutation. St000984The number of boxes below precisely one peak. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000082The number of elements smaller than a binary tree in Tamari order. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000224The sorting index of a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000471The sum of the ascent tops of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000616The inversion index of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001671Haglund's hag of a permutation. St001127The sum of the squares of the parts of a partition. St001684The reduced word complexity of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St000293The number of inversions of a binary word. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000915The Ore degree of a graph. 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. St000133The "bounce" of a permutation. St000327The number of cover relations in a poset. St000728The dimension of a set partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000156The Denert index of a permutation. St000225Difference between largest and smallest parts in a partition. St000259The diameter of a connected graph. St001280The number of parts of an integer partition that are at least two. St001651The Frankl number of a lattice. St001726The number of visible inversions of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001118The acyclic chromatic index of a graph. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000474Dyson's crank of a partition. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000209Maximum difference of elements in cycles. St000806The semiperimeter of the associated bargraph. 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 $c_{n−1}$. 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 $c_{n-1}$. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 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 St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001806The upper middle entry of a permutation. St001821The sorting index of a signed permutation. St001838The number of nonempty primitive factors of a binary word. St001875The number of simple modules with projective dimension at most 1. St000015The number of peaks of a Dyck path. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000439The position of the first down step of a Dyck path. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001060The distinguishing index of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001203We 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: 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. 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. St001530The depth of a Dyck path. St001658The total number of rook placements on a Ferrers board. St001733The number of weak left to right maxima of a Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St000005The bounce statistic of a Dyck path. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000456The monochromatic index of a connected graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000627The exponent of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000759The smallest missing part in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000922The minimal number such that all substrings of this length are unique. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000982The length of the longest constant subword. St000983The length of the longest alternating 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001210Gives 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. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical 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. St001238The 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001462The number of factors of a standard tableaux under concatenation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. 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). St001614The cyclic permutation representation number of a skew partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001809The index of the step at the first peak of maximal height in a Dyck path. St001884The number of borders of a binary word. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000010The length of the partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000292The number of ascents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000691The number of changes of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001219Number 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001480The number of simple summands of the module J^2/J^3. St001484The number of singletons of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St000545The number of parabolic double cosets with minimal element being the given permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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 $eA$. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001330The hat guessing number of a graph. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000101The cocharge of a semistandard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000958The number of Bruhat factorizations of a permutation. 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 $eA$. 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 $eA$. St000815The number of semistandard Young tableaux of partition weight of given shape. St001128The exponens consonantiae of a partition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama 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 $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000509The diagonal index (content) of a partition. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000744The length of the path to the largest entry in a standard Young tableau. St000933The number of multipartitions of sizes given by an integer partition. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. 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. 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). St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000928The sum of the coefficients of the character polynomial of an integer partition.