Your data matches 299 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001442
(load all 3 compositions to match this statistic)
Mapping
St001442: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 3 - 2
[2]
 => 1 = 3 - 2
[1,1]
 => 0 = 2 - 2
[3]
 => 1 = 3 - 2
[2,1]
 => 0 = 2 - 2
[1,1,1]
 => 1 = 3 - 2
[4]
 => 1 = 3 - 2
[3,1]
 => 0 = 2 - 2
[2,2]
 => 1 = 3 - 2
[2,1,1]
 => 1 = 3 - 2
[1,1,1,1]
 => 0 = 2 - 2
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
Matching statistic: St001159
(load all 65 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 1 = 3 - 2
[2]
 => [1,1,0,0,1,0]
 => 0 = 2 - 2
[1,1]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
[3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 2 - 2
[2,1]
 => [1,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 2 - 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 2 - 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
Description
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 11 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 2 = 3 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2 = 3 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => 2 = 3 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 2 = 3 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 2 - 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000542
(load all 14 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 2 = 3 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => 2 = 3 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 2 = 3 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 2 = 3 - 1
Description
The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000630
(load all 10 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 01 => 2 = 3 - 1
[2]
 => 100 => 101 => 1 = 2 - 1
[1,1]
 => 110 => 011 => 2 = 3 - 1
[3]
 => 1000 => 0101 => 2 = 3 - 1
[2,1]
 => 1010 => 1001 => 1 = 2 - 1
[1,1,1]
 => 1110 => 0111 => 2 = 3 - 1
[4]
 => 10000 => 10101 => 1 = 2 - 1
[3,1]
 => 10010 => 01101 => 2 = 3 - 1
[2,2]
 => 1100 => 1011 => 2 = 3 - 1
[2,1,1]
 => 10110 => 10001 => 1 = 2 - 1
[1,1,1,1]
 => 11110 => 01111 => 2 = 3 - 1
Description
The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [[1],[2]]
 => 2 = 3 - 1
[2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 3 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 2 = 3 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 1 = 2 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 3 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 3 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => 1 = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001024
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001201
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
Description
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
Matching statistic: St001257
(load all 30 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001257: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
Description
The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001390
(load all 13 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 2 = 3 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => 2 = 3 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 2 = 3 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 2 = 3 - 1
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
The following 289 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001733The number of weak left to right maxima of a Dyck path. St001884The number of borders of a binary word. St000153The number of adjacent cycles of a permutation. St000352The Elizalde-Pak rank of a permutation. St000390The number of runs of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001204Call 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001484The number of singletons of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001530The depth of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000061The number of nodes on the left branch of a binary tree. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000654The first descent of a permutation. St000733The row containing the largest entry of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000883The number of longest increasing subsequences of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001066The number of simple reflexive modules 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. 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. 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. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000023The number of inner peaks of a permutation. St000143The largest repeated part of a partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000237The number of small exceedances. St000257The number of distinct parts of a partition that occur at least twice. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000338The number of pixed points of a permutation. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000409The number of pitchforks in a binary tree. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000658The number of rises of length 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000665The number of rafts of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000877The depth of the binary word interpreted as a path. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000989The number of final rises of a permutation. 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. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001061The number of indices that are both descents and recoils of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001260The permanent of an alternating sign matrix. St001271The competition number of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001510The number of self-evacuating linear extensions of a finite poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001777The number of weak descents in an integer composition. St001850The number of Hecke atoms of a permutation. St001931The weak major index of an integer composition regarded as a word. St001948The number of augmented double ascents of a permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000823The number of unsplittable factors of the set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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 $eA$. St000876The number of factors in the Catalan decomposition of a binary word. 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 $eA$. St000706The product of the factorials of the multiplicities of an integer partition. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001413Half the length of the longest even length palindromic prefix of a binary word. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. 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. St000677The standardized bi-alternating inversion number of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. 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. 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St000050The depth or height of a binary tree. St000094The depth of an ordered tree. 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. St000444The length of the maximal rise of a Dyck path. St000458The number of permutations obtained by switching adjacencies or successions. St000494The number of inversions of distance at most 3 of a permutation. St000521The number of distinct subtrees of an ordered tree. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000638The number of up-down runs of a permutation. St000675The number of centered multitunnels of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. 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. St000797The stat`` of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St000842The breadth of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000981The length of the longest zigzag subpath. St000983The length of the longest alternating subword. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001415The length of the longest palindromic prefix of 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. St001437The flex of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001488The number of corners of a skew partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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. St001554The number of distinct nonempty subtrees of a binary tree. St000993The multiplicity of the largest part of an integer partition. St000239The number of small weak excedances. St000260The radius of a connected graph. St001153The number of blocks with even minimum in a set partition. St000075The orbit size of a standard tableau under promotion. St001060The distinguishing index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000302The determinant of the distance matrix of a connected graph. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000294The number of distinct factors of a binary word. St000438The position of the last up step in a Dyck path. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000656The number of cuts of a poset. St000668The least common multiple of the parts of the partition. St000680The Grundy value for Hackendot on posets. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000806The semiperimeter of the associated bargraph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000893The number of distinct diagonal sums of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000933The number of multipartitions of sizes given by an integer partition. St001074The number of inversions of the cyclic embedding of a permutation. St001118The acyclic chromatic index of a graph. St001050The number of terminal closers of a set partition. St001545The second Elser number of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000456The monochromatic index of a connected graph. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000782The indicator function of whether a given perfect matching is an L & P matching. St000873The aix statistic of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of 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. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000759The smallest missing part in an integer partition. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension 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. 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: St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001239The largest vector space dimension of the double dual of a simple module 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. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001471The magnitude of a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001814The number of partitions interlacing the given partition. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001399The distinguishing number of a poset.