Your data matches 459 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001571
(load all 4 compositions to match this statistic)
Mapping
St001571: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 2 = 3 - 1
[1,1]
 => 1 = 2 - 1
[3]
 => 3 = 4 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 1 = 2 - 1
[4]
 => 4 = 5 - 1
[3,1]
 => 2 = 3 - 1
[2,2]
 => 2 = 3 - 1
[2,1,1]
 => 1 = 2 - 1
[1,1,1,1]
 => 1 = 2 - 1
[5]
 => 5 = 6 - 1
[4,1]
 => 3 = 4 - 1
[3,2]
 => 2 = 3 - 1
[3,1,1]
 => 2 = 3 - 1
[2,2,1]
 => 1 = 2 - 1
[2,1,1,1]
 => 1 = 2 - 1
[1,1,1,1,1]
 => 1 = 2 - 1
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001933
Mapping
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 1 = 2 - 1
[1,1]
 => 2 = 3 - 1
[3]
 => 1 = 2 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 3 = 4 - 1
[4]
 => 1 = 2 - 1
[3,1]
 => 1 = 2 - 1
[2,2]
 => 2 = 3 - 1
[2,1,1]
 => 2 = 3 - 1
[1,1,1,1]
 => 4 = 5 - 1
[5]
 => 1 = 2 - 1
[4,1]
 => 1 = 2 - 1
[3,2]
 => 1 = 2 - 1
[3,1,1]
 => 2 = 3 - 1
[2,2,1]
 => 2 = 3 - 1
[2,1,1,1]
 => 3 = 4 - 1
[1,1,1,1,1]
 => 5 = 6 - 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001091
Mapping
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0 = 2 - 2
[2]
 => 0 = 2 - 2
[1,1]
 => 1 = 3 - 2
[3]
 => 0 = 2 - 2
[2,1]
 => 0 = 2 - 2
[1,1,1]
 => 2 = 4 - 2
[4]
 => 0 = 2 - 2
[3,1]
 => 0 = 2 - 2
[2,2]
 => 1 = 3 - 2
[2,1,1]
 => 1 = 3 - 2
[1,1,1,1]
 => 3 = 5 - 2
[5]
 => 0 = 2 - 2
[4,1]
 => 0 = 2 - 2
[3,2]
 => 0 = 2 - 2
[3,1,1]
 => 1 = 3 - 2
[2,2,1]
 => 1 = 3 - 2
[2,1,1,1]
 => 2 = 4 - 2
[1,1,1,1,1]
 => 4 = 6 - 2
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000120
(load all 13 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: 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
[2]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 6 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 4 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000392
(load all 9 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 1 = 2 - 1
[2]
 => 100 => 1 = 2 - 1
[1,1]
 => 110 => 2 = 3 - 1
[3]
 => 1000 => 1 = 2 - 1
[2,1]
 => 1010 => 1 = 2 - 1
[1,1,1]
 => 1110 => 3 = 4 - 1
[4]
 => 10000 => 1 = 2 - 1
[3,1]
 => 10010 => 1 = 2 - 1
[2,2]
 => 1100 => 2 = 3 - 1
[2,1,1]
 => 10110 => 2 = 3 - 1
[1,1,1,1]
 => 11110 => 4 = 5 - 1
[5]
 => 100000 => 1 = 2 - 1
[4,1]
 => 100010 => 1 = 2 - 1
[3,2]
 => 10100 => 1 = 2 - 1
[3,1,1]
 => 100110 => 2 = 3 - 1
[2,2,1]
 => 11010 => 2 = 3 - 1
[2,1,1,1]
 => 101110 => 3 = 4 - 1
[1,1,1,1,1]
 => 111110 => 5 = 6 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 10 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 1 = 2 - 1
[2]
 => 100 => 1 = 2 - 1
[1,1]
 => 110 => 2 = 3 - 1
[3]
 => 1000 => 1 = 2 - 1
[2,1]
 => 1010 => 1 = 2 - 1
[1,1,1]
 => 1110 => 3 = 4 - 1
[4]
 => 10000 => 1 = 2 - 1
[3,1]
 => 10010 => 1 = 2 - 1
[2,2]
 => 1100 => 2 = 3 - 1
[2,1,1]
 => 10110 => 2 = 3 - 1
[1,1,1,1]
 => 11110 => 4 = 5 - 1
[5]
 => 100000 => 1 = 2 - 1
[4,1]
 => 100010 => 1 = 2 - 1
[3,2]
 => 10100 => 1 = 2 - 1
[3,1,1]
 => 100110 => 2 = 3 - 1
[2,2,1]
 => 11010 => 2 = 3 - 1
[2,1,1,1]
 => 101110 => 3 = 4 - 1
[1,1,1,1,1]
 => 111110 => 5 = 6 - 1
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001498
(load all 10 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001498: 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
[2]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 6 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 4 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000381
(load all 9 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => [1,2] => 2
[2]
 => 100 => [1,3] => 3
[1,1]
 => 110 => [1,1,2] => 2
[3]
 => 1000 => [1,4] => 4
[2,1]
 => 1010 => [1,2,2] => 2
[1,1,1]
 => 1110 => [1,1,1,2] => 2
[4]
 => 10000 => [1,5] => 5
[3,1]
 => 10010 => [1,3,2] => 3
[2,2]
 => 1100 => [1,1,3] => 3
[2,1,1]
 => 10110 => [1,2,1,2] => 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => 2
[5]
 => 100000 => [1,6] => 6
[4,1]
 => 100010 => [1,4,2] => 4
[3,2]
 => 10100 => [1,2,3] => 3
[3,1,1]
 => 100110 => [1,3,1,2] => 3
[2,2,1]
 => 11010 => [1,1,2,2] => 2
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => 2
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => 2
Description
The largest part of an integer composition.
Matching statistic: St000451
(load all 21 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 2
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 3
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 4
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 4
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000982
(load all 11 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 11 => 2
[2]
 => 100 => 011 => 2
[1,1]
 => 110 => 111 => 3
[3]
 => 1000 => 0011 => 2
[2,1]
 => 1010 => 1101 => 2
[1,1,1]
 => 1110 => 1111 => 4
[4]
 => 10000 => 00011 => 3
[3,1]
 => 10010 => 01101 => 2
[2,2]
 => 1100 => 0111 => 3
[2,1,1]
 => 10110 => 11011 => 2
[1,1,1,1]
 => 11110 => 11111 => 5
[5]
 => 100000 => 000011 => 4
[4,1]
 => 100010 => 001101 => 2
[3,2]
 => 10100 => 11001 => 2
[3,1,1]
 => 100110 => 011011 => 2
[2,2,1]
 => 11010 => 11101 => 3
[2,1,1,1]
 => 101110 => 110111 => 3
[1,1,1,1,1]
 => 111110 => 111111 => 6
Description
The length of the longest constant subword.
The following 449 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000983The length of the longest alternating subword. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000876The number of factors in the Catalan decomposition of a binary word. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000996The number of exclusive left-to-right maxima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000147The largest part of an integer partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. 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 $c_{n-1}$. St000991The number of right-to-left minima of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001366The maximal multiplicity of a degree of a vertex of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001437The flex of a binary word. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000011The number of touch points (or returns) of a Dyck path. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000053The number of valleys of the Dyck path. St000056The decomposition (or block) number of a permutation. St000153The number of adjacent cycles of a permutation. St000160The multiplicity of the smallest part of a partition. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000216The absolute length of a permutation. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000354The number of recoils of a permutation. St000383The last part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000674The number of hills of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000730The maximal arc length of a set partition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000877The depth of the binary word interpreted as a path. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000999Number of indecomposable projective module with injective 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001050The number of terminal closers of a set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001246The maximal difference between two consecutive entries of a permutation. St001255The vector space 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001399The distinguishing number of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001726The number of visible inversions of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000074The number of special entries. St000143The largest repeated part of a partition. St000204The number of internal nodes of a binary tree. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000288The number of ones in a binary word. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000441The number of successions of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) 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. 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. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000931The number of occurrences of the pattern UUU in a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001411The number of patterns 321 or 3412 in a permutation. 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. St001584The area statistic between a Dyck path and its bounce path. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. 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. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001910The height of the middle non-run of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000740The last entry of a permutation. St000798The makl of a permutation. St000675The number of centered multitunnels of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000297The number of leading ones in a binary word. St001948The number of augmented double ascents of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000004The major index of a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000305The inverse major index of a permutation. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf 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. St000025The number of initial rises of a Dyck path. St000051The size of the left subtree of a binary 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. St000461The rix statistic of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000738The first entry in the last row of a standard tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000989The number of final rises of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001684The reduced word complexity of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001513The number of nested exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000993The multiplicity of the largest part of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. St000439The position of the first down step of a Dyck path. St000626The minimal period of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001570The minimal number of edges to add to make a graph Hamiltonian. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001730The number of times the path corresponding to a binary word crosses the base line. St001867The number of alignments of type EN of a signed permutation. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001864The number of excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St000406The number of occurrences of the pattern 3241 in a permutation. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000873The aix statistic of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001847The number of occurrences of the pattern 1432 in a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St000454The largest eigenvalue of a graph if it is integral. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000625The sum of the minimal distances to a greater element. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. 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 $eA$. St000014The number of parking functions supported by 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. St000144The pyramid weight of the Dyck path. St000294The number of distinct factors of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001400The total number of Littlewood-Richardson tableaux of given shape. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001814The number of partitions interlacing the given partition. St001556The number of inversions of the third entry of a permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000352The Elizalde-Pak rank of a permutation. St001645The pebbling number of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000264The girth of a graph, which is not a tree. St000839The largest opener of a set partition. St001545The second Elser number of a connected graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000090The variation of a composition. St000091The descent variation of a composition. St000230Sum of the minimal elements of the blocks of a set partition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St001151The number of blocks with odd minimum. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000455The second largest eigenvalue of a graph if it is integral. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000735The last entry on the main diagonal of a standard tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000307The number of rowmotion orbits of a poset. St000418The number of Dyck paths that are weakly below a Dyck path. St000438The position of the last up step in a Dyck path. St000477The weight of a partition according to Alladi. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000770The major index of an integer partition when read from bottom to top. St000806The semiperimeter of the associated bargraph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000981The length of the longest zigzag subpath. St001118The acyclic chromatic index of a graph. St001531Number of partial orders contained in the poset determined by the Dyck path. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001959The product of the heights of the peaks of a Dyck path. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. 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 $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001095The number of non-isomorphic posets with precisely one further covering relation. St001520The number of strict 3-descents.