Your data matches 99 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001129
(load all 6 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St001129: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 1 = 0 + 1
[2]
 => [1,1]
 => 1 = 0 + 1
[1,1]
 => [2]
 => 4 = 3 + 1
[2,1]
 => [1,1,1]
 => 1 = 0 + 1
Description
The product of the squares of the parts of a partition.
Matching statistic: St001562
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001562: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 1 = 0 + 1
[2]
 => [2]
 => 1 = 0 + 1
[1,1]
 => [1,1]
 => 4 = 3 + 1
[2,1]
 => [3]
 => 1 = 0 + 1
Description
The value of the complete homogeneous symmetric function evaluated at 1. The statistic is $h_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$, where $\lambda$ has $k$ parts.
Matching statistic: St001563
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001563: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 1 = 0 + 1
[2]
 => [2]
 => 1 = 0 + 1
[1,1]
 => [1,1]
 => 4 = 3 + 1
[2,1]
 => [3]
 => 1 = 0 + 1
Description
The value of the power-sum symmetric function evaluated at 1. The statistic is $p_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$, where $\lambda$ has $k$ parts.
Matching statistic: St000226
(load all 24 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000226: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
Description
The convexity of a permutation. It is given by the maximal value of $2x_i-x_{i-1}-x_{i+1}$ over all $i \in \{2,\ldots,n-1\}$.
Matching statistic: St000296
(load all 16 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 01 => 0
[2]
 => 100 => 001 => 0
[1,1]
 => 110 => 101 => 3
[2,1]
 => 1010 => 0101 => 0
Description
The length of the symmetric border of a binary word. The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix. The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St001519
(load all 26 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001519: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
Description
The pinnacle sum of a permutation. This is, the sum of the pinnacles (peak values) given by $$\sum_{i \text{ peak of } \sigma} \sigma(i).$$
Matching statistic: St000712
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000712: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => []
 => 1 = 0 + 1
[2]
 => [2]
 => []
 => 1 = 0 + 1
[1,1]
 => [1,1]
 => [1]
 => 4 = 3 + 1
[2,1]
 => [3]
 => []
 => 1 = 0 + 1
Description
The number of semistandard Young tableau of given shape, with entries at most 4. This is also the dimension of the corresponding irreducible representation of $GL_4$.
Matching statistic: St001885
(load all 8 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001885: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 00 => 1 = 0 + 1
[2]
 => 100 => 010 => 1 = 0 + 1
[1,1]
 => 110 => 001 => 4 = 3 + 1
[2,1]
 => 1010 => 0000 => 1 = 0 + 1
Description
The number of binary words with the same proper border set. The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$. For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
Matching statistic: St000950
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000950: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0]
 => 2 = 0 + 2
[2]
 => [2]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,1]
 => [1,1]
 => [1,1,0,0]
 => 5 = 3 + 2
[2,1]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
Description
Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.
Matching statistic: St000964
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000964: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 0 + 3
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 0 + 3
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 6 = 3 + 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 0 + 3
Description
Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra.
The following 89 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000027The major index of a Dyck path. St000089The absolute variation of a composition. St000117The number of centered tunnels of a Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000297The number of leading ones in a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000357The number of occurrences of the pattern 12-3. St000391The sum of the positions of the ones in a binary word. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000637The length of the longest cycle in a graph. St000674The number of hills of a Dyck path. 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. St000709The number of occurrences of 14-2-3 or 14-3-2. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001478The number of nowhere zero 4-flows of a graph. St001910The height of the middle non-run of a Dyck path. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000154The sum of the descent bottoms of a permutation. St000238The number of indices that are not small weak excedances. St000266The number of spanning subgraphs of a graph with the same connected components. St000326The position of the first one in a binary word after appending a 1 at the end. St000402Half the size of the symmetry class of a permutation. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000463The number of admissible inversions of a permutation. St000501The size of the first part in the decomposition of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000692Babson and Steingrímsson's statistic of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000958The number of Bruhat factorizations of a permutation. 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 $c_{n−1}$. St001081The number of minimal length factorizations of a permutation into star transpositions. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001468The smallest fixpoint of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001735The number of permutations with the same set of runs. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001807The lower middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000156The Denert index of a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000418The number of Dyck paths that are weakly below a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000762The sum of the positions of the weak records of an integer composition. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000978The sum of the positions of double down-steps of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001964The interval resolution global dimension of a poset. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$.