Your data matches 804 different statistics following compositions of up to 3 maps.
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St000533: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 2
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
Description
The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Mp00202: Integer partitions first row removalInteger partitions
St000547: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 0 = 1 - 1
[2]
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> 0 = 1 - 1
[3]
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4]
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> 0 = 1 - 1
[2,2]
=> [2]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[5]
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> 0 = 1 - 1
[3,2]
=> [2]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 3 - 1
Description
The number of even non-empty partial sums of an integer partition.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000092: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
Description
The number of outer peaks of a permutation. An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St000628: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 1
[1,1]
=> 110 => 011 => 1
[3]
=> 1000 => 0001 => 1
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 1
[4]
=> 10000 => 00001 => 1
[3,1]
=> 10010 => 00011 => 2
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 2
[1,1,1,1]
=> 11110 => 01111 => 1
[5]
=> 100000 => 000001 => 1
[4,1]
=> 100010 => 000011 => 2
[3,2]
=> 10100 => 00011 => 2
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 2
[2,1,1,1]
=> 101110 => 001111 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Mp00095: Integer partitions to binary wordBinary words
Mp00261: Binary words Burrows-WheelerBinary words
St000875: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 10 => 1
[2]
=> 100 => 100 => 1
[1,1]
=> 110 => 110 => 1
[3]
=> 1000 => 1000 => 1
[2,1]
=> 1010 => 1100 => 2
[1,1,1]
=> 1110 => 1110 => 1
[4]
=> 10000 => 10000 => 1
[3,1]
=> 10010 => 11000 => 2
[2,2]
=> 1100 => 1010 => 2
[2,1,1]
=> 10110 => 11100 => 2
[1,1,1,1]
=> 11110 => 11110 => 1
[5]
=> 100000 => 100000 => 1
[4,1]
=> 100010 => 101000 => 2
[3,2]
=> 10100 => 11000 => 2
[3,1,1]
=> 100110 => 110010 => 3
[2,2,1]
=> 11010 => 11100 => 2
[2,1,1,1]
=> 101110 => 111010 => 2
[1,1,1,1,1]
=> 111110 => 111110 => 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001202: 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]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
Description
Call 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. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra. The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St001420: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 1
[1,1]
=> 110 => 011 => 1
[3]
=> 1000 => 0001 => 1
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 1
[4]
=> 10000 => 00001 => 1
[3,1]
=> 10010 => 00011 => 2
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 2
[1,1,1,1]
=> 11110 => 01111 => 1
[5]
=> 100000 => 000001 => 1
[4,1]
=> 100010 => 000011 => 2
[3,2]
=> 10100 => 00011 => 2
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 2
[2,1,1,1]
=> 101110 => 001111 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 1
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St001514: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Matching statistic: St001778
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St001778: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
Description
The largest greatest common divisor of an element and its image in a permutation.
The following 794 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000242The number of indices that are not cyclical small weak excedances. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000647The number of big descents of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001469The holeyness of a permutation. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St000010The length of the partition. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000201The number of leaf nodes in a binary tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000273The domination number of a graph. St000288The number of ones in a binary word. St000308The height of the tree associated to a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000354The number of recoils of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000413The number of ordered trees with the same underlying unordered tree. St000470The number of runs in a permutation. St000527The width of the poset. St000619The number of cyclic descents of a permutation. St000657The smallest part of an integer composition. St000662The staircase size of the code of a permutation. St000767The number of runs in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000789The number of crossing-similar perfect matchings of a perfect matching. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000906The length of the shortest maximal chain in a poset. St000956The maximal displacement of a permutation. St000982The length of the longest constant subword. St000996The number of exclusive left-to-right maxima of a permutation. St001029The size of the core of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001116The game chromatic number of a graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001220The width of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001339The irredundance number of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001432The order dimension of the partition. St001486The number of corners of the ribbon associated with an integer composition. St001487The number of inner corners of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001494The Alon-Tarsi number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001829The common independence number of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001924The number of cells in an integer partition whose arm and leg length coincide. St001928The number of non-overlapping descents in a permutation. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000057The Shynar inversion number of a standard tableau. St000149The number of cells of the partition whose leg is zero and arm is odd. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000204The number of internal nodes of a binary tree. St000245The number of ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000534The number of 2-rises of a permutation. St000536The pathwidth of a graph. St000552The number of cut vertices of a graph. St000663The number of right floats of a permutation. St000731The number of double exceedences of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St000884The number of isolated descents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 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. St001214The aft of an integer partition. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001471The magnitude of a Dyck path. St001537The number of cyclic crossings of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001781The interlacing number of a set partition. St001792The arboricity of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000291The number of descents of a binary word. St000353The number of inner valleys of a permutation. St000538The number of even inversions of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001388The number of non-attacking neighbors of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000060The greater neighbor of the maximum. St000568The hook number of a binary tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000886The number of permutations with the same antidiagonal sums. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001597The Frobenius rank of a skew partition. St000238The number of indices that are not small weak excedances. St000497The lcb statistic of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000670The reversal length of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000893The number of distinct diagonal sums of an alternating sign matrix. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000402Half the size of the symmetry class of a permutation. St001405The number of bonds in a permutation. St001569The maximal modular displacement of a permutation. St001948The number of augmented double ascents of a permutation. St000137The Grundy value of an integer partition. St000365The number of double ascents of a permutation. St001481The minimal height of a peak of a Dyck path. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000702The number of weak deficiencies of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001118The acyclic chromatic index of a graph. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000632The jump number of the poset. St000989The number of final rises of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $eA$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000640The rank of the largest boolean interval in a poset. St001260The permanent of an alternating sign matrix. St001435The number of missing boxes in the first row. St001863The number of weak excedances of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000015The number of peaks of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000346The number of coarsenings of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000759The smallest missing part in an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 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: St001250The number of parts of a partition that are not congruent 0 modulo 3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001488The number of corners of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001530The depth of a Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000006The dinv of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000058The order of a permutation. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000397The Strahler number of a rooted tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000485The length of the longest cycle of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000522The number of 1-protected nodes of a rooted tree. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000676The number of odd rises of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000780The size of the orbit under rotation of a perfect matching. St000808The number of up steps of the associated bargraph. St000831The number of indices that are either descents or recoils. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000839The largest opener of a set partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000903The number of different parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000945The number of matchings in the dihedral orbit of a perfect matching. St000988The orbit size of a permutation under Foata's bijection. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001052The length of the exterior of a permutation. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in 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. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001372The length of a longest cyclic run of ones of a binary word. St001375The pancake length of a permutation. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001589The nesting number of a perfect matching. St001624The breadth of a lattice. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001864The number of excedances of a signed permutation. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001955The number of natural descents for set-valued two row standard Young tableaux. St001811The Castelnuovo-Mumford regularity of a permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000075The orbit size of a standard tableau under promotion. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000260The radius of a connected graph. St000331The number of upper interactions of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000899The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001114The number of odd descents of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001470The cyclic holeyness of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001820The size of the image of the pop stack sorting operator. St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000648The number of 2-excedences of a permutation. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000703The number of deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000942The number of critical left to right maxima of the parking functions. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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 $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001438The number of missing boxes of a skew partition. St001667The maximal size of a pair of weak twins for a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001846The number of elements which do not have a complement in the lattice. St001866The nesting alignments of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001964The interval resolution global dimension of a poset. St000451The length of the longest pattern of the form k 1 2. St000031The number of cycles in the cycle decomposition of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000061The number of nodes on the left branch of a binary tree. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000327The number of cover relations in a poset. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000487The length of the shortest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000530The number of permutations with the same descent word as the given permutation. St000654The first descent of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000729The minimal arc length of a set partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St000823The number of unsplittable factors of the set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000925The number of topologically connected components of a set partition. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001062The maximal size of a block of a set partition. St001128The exponens consonantiae of a partition. St001246The maximal difference between two consecutive entries of a permutation. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001437The flex of a binary word. St001516The number of cyclic bonds of a permutation. St001531Number of partial orders contained in the poset determined by the Dyck path. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001637The number of (upper) dissectors of a poset. St001808The box weight or horizontal decoration of a Dyck path. St001885The number of binary words with the same proper border set. St001959The product of the heights of the peaks of a Dyck path. St000077The number of boxed and circled entries. St000259The diameter of a connected graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000842The breadth of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000068The number of minimal elements in a poset. St000679The pruning number of an ordered tree. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St000026The position of the first return 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. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000439The position of the first down step of a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000644The number of graphs with given frequency partition. St000667The greatest common divisor of the parts of the partition. St000674The number of hills of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension 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. St001060The distinguishing index of a graph. St001127The sum of the squares of the parts of a partition. St001162The minimum jump of a permutation. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001344The neighbouring number of a permutation. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001400The total number of Littlewood-Richardson tableaux of given shape. 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. St001419The length of the longest palindromic factor beginning with a one of 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. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000090The variation of a composition. St000091The descent variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000815The number of semistandard Young tableaux of partition weight of given shape. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000741The Colin de Verdière graph invariant. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001490The number of connected components of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000770The major index of an integer partition when read from bottom to top. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St000022The number of fixed points of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000862The number of parts of the shifted shape of a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001875The number of simple modules with projective dimension at most 1. St000045The number of linear extensions of a binary tree. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000253The crossing number of a set partition. St000284The Plancherel distribution on integer partitions. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000486The number of cycles of length at least 3 of a permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000563The number of overlapping pairs of blocks of a set partition. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000694The number of affine bounded permutations that project to a given permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001399The distinguishing number of a poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001545The second Elser number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000039The number of crossings of a permutation. St000084The number of subtrees. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000251The number of nonsingleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. 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. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001151The number of blocks with odd minimum. St001301The first Betti number of the order complex associated with the poset. 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. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000824The sum of the number of descents and the number of recoils of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000782The indicator function of whether a given perfect matching is an L & P matching. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St000874The position of the last double rise in a Dyck path. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000997The even-odd crank of an integer partition. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000102The charge of a semistandard tableau.