Your data matches 308 different statistics following compositions of up to 3 maps.
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Matching statistic: St001213
(load all 40 compositions to match this statistic)
Mapping
St001213: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 4
[1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => 6
[1,0,1,1,0,0]
 => 5
[1,1,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => 6
[1,1,1,0,0,0]
 => 4
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St000438
(load all 64 compositions to match this statistic)
Mapping
St000438: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => 5 = 6 - 1
[1,0,1,1,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0]
 => 5 = 6 - 1
[1,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,1,0,0,0]
 => 3 = 4 - 1
Description
The position of the last up step in a Dyck path.
Matching statistic: St000395
(load all 16 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 6
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001034
(load all 18 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 6
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000734
(load all 13 compositions to match this statistic)
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [[1,3],[2,4]]
 => 3 = 4 - 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 5 = 6 - 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 4 = 5 - 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 5 = 6 - 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 4 = 5 - 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3 = 4 - 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000841
(load all 17 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000841: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [(1,2),(3,4)]
 => 3 = 4 - 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 5 = 6 - 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 4 = 5 - 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 5 = 6 - 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 4 = 5 - 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3 = 4 - 1
Description
The largest opener of a perfect matching. An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint).
Matching statistic: St000029
(load all 11 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 6
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
Description
The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000070
(load all 3 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => ([],2)
 => 4
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => 3
[1,0,1,0,1,0]
 => [2,3,1] => ([(1,2)],3)
 => 6
[1,0,1,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 5
[1,1,0,0,1,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 5
[1,1,0,1,0,0]
 => [3,1,2] => ([(1,2)],3)
 => 6
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
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: St000176
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000176: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [[1,3],[2,4]]
 => [[2,2,0,0],[2,1,0],[1,1],[1]]
 => 3
[1,1,0,0]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 4
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => 4
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => 5
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 5
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
 => 6
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 6
Description
The total number of tiles in the Gelfand-Tsetlin pattern. The tiling of a Gelfand-Tsetlin pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belongs to exactly one tile.
Matching statistic: St000189
(load all 5 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6
Description
The number of elements in the poset.
The following 298 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000197The number of entries equal to positive one in the alternating sign matrix. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000301The number of facets of the stable set polytope of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000656The number of cuts of a poset. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. 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. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001717The largest size of an interval in a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000505The biggest entry in the block containing the 1. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000641The number of non-empty boolean intervals in a poset. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000806The semiperimeter of the associated bargraph. St000833The comajor index of a permutation. St000839The largest opener of a set partition. St000874The position of the last double rise in a Dyck path. St000971The smallest closer of a set partition. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001497The position of the largest weak excedence of a permutation. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001759The Rajchgot index of a permutation. St000074The number of special entries. St000339The maf index of a permutation. St000446The disorder of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001375The pancake length of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001956The comajor index for set-valued two-row standard Young tableaux. St000246The number of non-inversions of a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St001077The prefix exchange distance of a permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000012The area of a Dyck path. St000030The sum of the descent differences of a permutations. St000054The first entry of the permutation. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000180The number of chains of a poset. St000224The sorting index of a permutation. St000231Sum of the maximal elements of the blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000288The number of ones in a binary word. St000300The number of independent sets of vertices of a graph. St000305The inverse major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000393The number of strictly increasing runs in a binary word. St000400The path length of an ordered tree. St000443The number of long tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000626The minimal period of a binary word. St000673The number of non-fixed points of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000728The dimension of a set partition. St000794The mak of a permutation. St000797The stat`` of a permutation. St000809The reduced reflection length of the permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868The aid statistic in the sense of Shareshian-Wachs. St000915The Ore degree of a graph. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. 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. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. 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. 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. St001437The flex of a binary word. St001458The rank of the adjacency matrix of a graph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001616The number of neutral elements in a lattice. St001725The harmonious chromatic number of a graph. St001726The number of visible inversions of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001869The maximum cut size of a graph. St001894The depth of a signed permutation. St000024The number of double up and double down steps of a Dyck path. St000060The greater neighbor of the maximum. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000108The number of partitions contained in the given partition. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000156The Denert index of a permutation. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000211The rank of the set partition. St000271The chromatic index of a graph. St000304The load of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000332The positive inversions of an alternating sign matrix. St000391The sum of the positions of the ones in a binary word. St000459The hook length of the base cell of a partition. St000495The number of inversions of distance at most 2 of a permutation. St000503The maximal difference between two elements in a common block. St000507The number of ascents of a standard tableau. St000567The sum of the products of all pairs of parts. St000619The number of cyclic descents of a permutation. St000625The sum of the minimal distances to a greater element. St000639The number of relations in a poset. St000653The last descent of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000798The makl of a permutation. St000831The number of indices that are either descents or recoils. St000867The sum of the hook lengths in the first row of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000956The maximal displacement of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. 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. 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. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001388The number of non-attacking neighbors of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001415The length of the longest palindromic prefix of a binary word. St001428The number of B-inversions of a signed permutation. St001468The smallest fixpoint of a permutation. St001480The number of simple summands of the module J^2/J^3. 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. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001671Haglund's hag of a permutation. St001721The degree of a binary word. St001760The number of prefix or suffix reversals needed to sort a permutation. St001807The lower middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001883The mutual visibility number of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001978The codimension of the alternating sign matrix variety. St000006The dinv of a Dyck path. St000009The charge of a standard tableau. St000058The order of a permutation. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000222The number of alignments in the permutation. St000290The major index of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000485The length of the longest cycle of a permutation. St000499The rcb statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000539The number of odd inversions of a permutation. St000572The dimension exponent of a set partition. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000638The number of up-down runs of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000670The reversal length of a permutation. St000691The number of changes of a binary word. St000747A variant of the major index of a set partition. St000836The number of descents of distance 2 of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001346The number of parking functions that give the same permutation. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001806The upper middle entry of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000091The descent variation of a composition. St000136The dinv of a parking function. St000154The sum of the descent bottoms of a permutation. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000289The decimal representation of a binary word. St000357The number of occurrences of the pattern 12-3. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000647The number of big descents of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000711The number of big exceedences of a permutation. St000719The number of alignments in a perfect matching. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000946The sum of the skew hook positions in a Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001094The depth index of a set partition. St001161The major index north count of a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001535The number of cyclic alignments of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001874Lusztig's a-function for the symmetric group. St001911A descent variant minus the number of inversions. St001930The weak major index of a binary word. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001644The dimension of a graph. St000770The major index of an integer partition when read from bottom to top. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001812The biclique partition number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000307The number of rowmotion orbits of a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001626The number of maximal proper sublattices of a lattice. St001645The pebbling number of a connected graph. St001623The number of doubly irreducible elements of a lattice. St000177The number of free tiles in the pattern. St001845The number of join irreducibles minus the rank of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001625The Möbius invariant of a lattice.