Your data matches 107 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00322: Integer partitions Loehr-WarringtonInteger 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]
=> [1,1]
=> 4 = 3 + 1
[1,1]
=> [2]
=> 1 = 0 + 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.
Mp00322: Integer partitions Loehr-WarringtonInteger 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]
=> [1,1]
=> 4 = 3 + 1
[1,1]
=> [2]
=> 1 = 0 + 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.
Mp00095: Integer partitions to binary wordBinary words
Mp00096: Binary words Foata bijectionBinary words
St000296: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 10 => 0
[2]
=> 100 => 010 => 3
[1,1]
=> 110 => 110 => 0
[2,1]
=> 1010 => 1100 => 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: St000674
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
Description
The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.
Mp00095: Integer partitions to binary wordBinary words
Mp00096: Binary words Foata bijectionBinary words
St001371: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 10 => 0
[2]
=> 100 => 010 => 3
[1,1]
=> 110 => 110 => 0
[2,1]
=> 1010 => 1100 => 0
Description
The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
St001932: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
Description
The 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. Let $D$ be a Dyck path, and let $P$ be the noncrossing set partition obtained by applying [[Mp00138]]. For each pair of singleton blocks $\{a\}, \{b\}$, let $P'$ be the set partition obtained from $P$ by merging the two blocks. This statistic enumerates the number of (unordered) pairs of singleton blocks such that $P'$ is noncrossing.
Matching statistic: St000712
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
Mp00202: Integer partitions first row removalInteger 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]
=> [1,1]
=> [1]
=> 4 = 3 + 1
[1,1]
=> [2]
=> []
=> 1 = 0 + 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 $GL_4$.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
St000968: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
Description
We 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 $c_{n−1}$. Then we calculate the dominant dimension of that CNakayama algebra.
Mp00044: Integer partitions conjugateInteger partitions
Mp00323: Integer partitions Loehr-Warrington inverseInteger partitions
St001129: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [1]
=> 1 = 0 + 1
[2]
=> [1,1]
=> [2]
=> 4 = 3 + 1
[1,1]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[2,1]
=> [2,1]
=> [1,1,1]
=> 1 = 0 + 1
Description
The product of the squares of the parts of a partition.
Matching statistic: St001800
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
St001800: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
Description
The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$. Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by projecting the Catalan path onto the $x,y$-plane and the $y,z$-plane, respectively. For a given Dyck path $D$ this is the number of Catalan paths $C$ such that $D_{xy}(C) = D_{yz}(C) = D$. If $D$ is of semilength $n$, $r_i(D)$ denotes the number of downsteps between the $i$-th and $(i+1)$-st upstep, and $s_i(D)$ denotes the number of upsteps between the $i$-th and $(i+1)$-st downstep, then this number is given by $\prod\limits_{i=1}^{n-1} \binom{r_i(D) + s_i(D)}{r_i(D)}$.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001807The lower middle entry of a permutation. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000008The major index of the composition. St000022The number of fixed points of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000210Minimum over maximum difference of elements in cycles. St000221The number of strong fixed points of a permutation. St000226The convexity of a permutation. St000241The number of cyclical small 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. 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. St000463The number of admissible inversions of a permutation. St000498The lcs statistic of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000616The inversion index of a permutation. 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. 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. 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. St001077The prefix exchange distance of a permutation. 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001375The pancake length of a permutation. St001379The number of inversions plus the major index of a permutation. St001519The pinnacle sum of a permutation. St001910The height of the middle non-run of a Dyck path. St001911A descent variant minus the number of inversions. St001931The weak major index of an integer composition regarded as a word. St000054The first entry of the permutation. St000242The number of indices that are not cyclical small weak excedances. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000391The sum of the positions of the ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000487The length of the shortest cycle 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. St000763The sum of the positions of the strong records of an integer composition. St000809The reduced reflection length of the permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000958The number of Bruhat factorizations of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001081The number of minimal length factorizations of a permutation into star transpositions. St001468The smallest fixpoint of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001671Haglund's hag of a permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001735The number of permutations with the same set of runs. St001806The upper middle entry of a permutation. St001838The number of nonempty primitive factors of a binary word. St001885The number of binary words with the same proper border set. St000032The number of elements smaller than the given Dyck path in the Tamari Order. 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. St000946The sum of the skew hook positions in a Dyck path. St001415The length of the longest palindromic prefix of a binary word. St000762The sum of the positions of the weak records of an integer composition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000976The sum of the positions of double up-steps of a Dyck path. 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. St000694The number of affine bounded permutations that project to a given permutation. St001964The interval resolution global dimension of a poset. St000260The radius of a connected graph. St000545The number of parabolic double cosets with minimal element being the given permutation. 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. 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 $eA$. St000524The number of posets with the same order polynomial.