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Your data matches 958 different statistics following compositions of up to 3 maps.
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Matching statistic: St000944
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(load all 5 compositions to match this statistic)
St000944: 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]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 1 = 2 - 1
[3,1]
=> 0 = 1 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 0 = 1 - 1
[1,1,1,1]
=> 0 = 1 - 1
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St000920
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000920: 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]
=> [1,1,0,0,1,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001913
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 1
[3]
=> [2,1]
=> 2
[2,1]
=> [2,1]
=> 2
[1,1,1]
=> [3]
=> 1
[4]
=> [3,1]
=> 2
[3,1]
=> [2,2]
=> 1
[2,2]
=> [2,1,1]
=> 1
[2,1,1]
=> [3,1]
=> 2
[1,1,1,1]
=> [4]
=> 1
Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
Matching statistic: St000660
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 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]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001414
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 0 = 1 - 1
[2]
=> 100 => 0 = 1 - 1
[1,1]
=> 110 => 0 = 1 - 1
[3]
=> 1000 => 0 = 1 - 1
[2,1]
=> 1010 => 1 = 2 - 1
[1,1,1]
=> 1110 => 1 = 2 - 1
[4]
=> 10000 => 0 = 1 - 1
[3,1]
=> 10010 => 0 = 1 - 1
[2,2]
=> 1100 => 0 = 1 - 1
[2,1,1]
=> 10110 => 1 = 2 - 1
[1,1,1,1]
=> 11110 => 1 = 2 - 1
Description
Half the length of the longest odd length palindromic prefix of a binary word.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
Matching statistic: St001423
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St001423: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001423: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 0 = 1 - 1
[2]
=> 100 => 0 = 1 - 1
[1,1]
=> 110 => 0 = 1 - 1
[3]
=> 1000 => 1 = 2 - 1
[2,1]
=> 1010 => 0 = 1 - 1
[1,1,1]
=> 1110 => 1 = 2 - 1
[4]
=> 10000 => 1 = 2 - 1
[3,1]
=> 10010 => 0 = 1 - 1
[2,2]
=> 1100 => 0 = 1 - 1
[2,1,1]
=> 10110 => 0 = 1 - 1
[1,1,1,1]
=> 11110 => 1 = 2 - 1
Description
The number of distinct cubes in a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uuu$ is a factor of the word.
Matching statistic: St001524
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 0 = 1 - 1
[2]
=> 100 => 0 = 1 - 1
[1,1]
=> 110 => 0 = 1 - 1
[3]
=> 1000 => 1 = 2 - 1
[2,1]
=> 1010 => 0 = 1 - 1
[1,1,1]
=> 1110 => 1 = 2 - 1
[4]
=> 10000 => 1 = 2 - 1
[3,1]
=> 10010 => 0 = 1 - 1
[2,2]
=> 1100 => 0 = 1 - 1
[2,1,1]
=> 10110 => 0 = 1 - 1
[1,1,1,1]
=> 11110 => 1 = 2 - 1
Description
The degree of symmetry of a binary word.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Matching statistic: St000024
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(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000201
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000329
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
The following 948 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000396The register function (or Horton-Strahler number) of a binary tree. St000442The maximal area to the right of an up step of a Dyck path. St000527The width of the poset. St000669The number of permutations obtained by switching ascents or descents of size 2. St000889The number of alternating sign matrices with the same antidiagonal sums. St000922The minimal number such that all substrings of this length are unique. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001597The Frobenius rank of a skew partition. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001778The largest greatest common divisor of an element and its image in a permutation. 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). St000013The height of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. 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. St000443The number of long tunnels of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000650The number of 3-rises of a permutation. St000711The number of big exceedences of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000893The number of distinct diagonal sums of an alternating sign matrix. St001007Number of simple modules with projective dimension 1 in 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. St001022Number of simple modules with projective dimension 3 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. 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. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001224Let X be the direct sum of all simple modules of 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. St001394The genus of a permutation. St001403The number of vertical separators in a permutation. St001530The depth of a Dyck path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001673The degree of asymmetry of an integer composition. St001728The number of invisible descents of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000001The number of reduced words for a permutation. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000079The number of alternating sign matrices for a given Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000120The number of left tunnels of a Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000245The number of ascents of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000306The bounce count of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000413The number of ordered trees with the same underlying unordered tree. St000417The size of the automorphism group of the ordered tree. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000537The cutwidth of a graph. St000539The number of odd inversions 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. St000547The number of even non-empty partial sums of an integer partition. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000655The length of the minimal rise of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000675The number of centered multitunnels of a Dyck path. St000679The pruning number of an ordered tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000701The protection number of a binary tree. St000703The number of deficiencies of a permutation. St000710The number of big 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. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000809The reduced reflection length of the permutation. St000820The number of compositions obtained by rotating the composition. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000886The number of permutations with the same antidiagonal sums. St000897The number of different multiplicities of parts of an integer partition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000988The orbit size of a permutation under Foata's bijection. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima 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. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001052The length of the exterior of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001081The number of minimal length factorizations of a permutation into star transpositions. St001096The size of the overlap set of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001152The number of pairs with even minimum in a perfect matching. St001162The minimum jump of a permutation. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001267The length of the Lyndon factorization of the binary word. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001270The bandwidth of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001281The normalized isoperimetric number of a graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001405The number of bonds in a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of 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. St001437The flex of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001569The maximal modular displacement of a permutation. St001644The dimension of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a 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. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001962The proper pathwidth of a graph. St000015The number of peaks of a Dyck path. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000117The number of centered tunnels of a Dyck path. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000133The "bounce" of a permutation. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St000256The number of parts from which one can substract 2 and still get an integer partition. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000308The height of the tree associated to a permutation. St000313The number of degree 2 vertices of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. 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. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000397The Strahler number of a rooted tree. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000542The number of left-to-right-minima of a permutation. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000676The number of odd rises of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000702The number of weak deficiencies of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000731The number of double exceedences of a permutation. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000808The number of up steps of the associated bargraph. St000824The sum of the number of descents and the number of recoils of a permutation. St000836The number of descents of distance 2 of a permutation. St000842The breadth of a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000982The length of the longest constant subword. St000989The number of final rises 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. St001033The normalized area of the parallelogram polyomino associated with 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. 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. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 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 $eA$. 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 $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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:
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 $eA$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001335The cardinality of a minimal cycle-isolating set of a graph. St001377The major index minus the number of inversions of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001471The magnitude of a Dyck path. St001513The number of nested exceedences of a permutation. 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. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001584The area statistic between a Dyck path and its bounce path. St001589The nesting number of a perfect matching. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001638The book thickness of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001642The Prague dimension of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. 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. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. 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. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000094The depth of an ordered tree. St000187The determinant of an alternating sign matrix. St000521The number of distinct subtrees of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001557The number of inversions of the second entry of a permutation. St000402Half the size of the symmetry class of a permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St001128The exponens consonantiae of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001948The number of augmented double ascents of a permutation. St000654The first descent of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001964The interval resolution global dimension of a poset. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000910The number of maximal chains of minimal length in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001768The number of reduced words of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001823The Stasinski-Voll length of a signed permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001586The number of odd parts smaller than the largest even part in an integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001846The number of elements which do not have a complement in the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000990The first ascent of a permutation. St001487The number of inner corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001114The number of odd descents of a permutation. St001488The number of corners of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000147The largest part of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. 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. St000531The leading coefficient of the rook polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000759The smallest missing part in an integer partition. St000935The number of ordered refinements of an integer partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001389The number of partitions of the same length below the given integer partition. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001809The index of the step at the first peak of maximal height in 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. St000010The length of the partition. St000014The number of parking functions supported by a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000038The product of the heights of the descending steps of a Dyck path. St000048The multinomial of the parts of a partition. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000060The greater neighbor of the maximum. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000160The multiplicity of the smallest part of a partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000340The number of non-final maximal constant sub-paths of length greater than one. St000346The number of coarsenings of a partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000385The number of vertices with out-degree 1 in a binary tree. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000545The number of parabolic double cosets with minimal element being the given permutation. St000548The number of different non-empty partial sums of an integer partition. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000657The smallest part of an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000700The protection number of an ordered tree. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. 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. 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. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000847The number of standard Young tableaux whose descent set is the binary word. St000868The aid statistic in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000939The number of characters of the symmetric group whose value on the partition is positive. St000952Gives the number of irreducible factors 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 $c_{n−1}$. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000993The multiplicity of the largest part of an integer partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001285The number of primes in the column sums of the two line notation of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001432The order dimension of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. 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. St001486The number of corners of the ribbon associated with an integer composition. 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001652The length of a longest interval of consecutive numbers. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001946The number of descents in a parking function. St001955The number of natural descents for set-valued two row standard Young tableaux. St001956The comajor index for set-valued two-row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001822The number of alignments of a signed permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000693The modular (standard) major index of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001624The breadth of a lattice. 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. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 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. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs 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. St001611The number of multiset partitions such that the multiplicities of elements 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. St001645The pebbling number of a connected graph. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St000022The number of fixed points of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. 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. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000546The number of global descents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001722The number of minimal chains with small intervals between a binary word and the top element. St001808The box weight or horizontal decoration of a Dyck path. St000045The number of linear extensions of a binary tree. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. 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. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000296The length of the symmetric border of a binary word. St000327The number of cover relations in a poset. St000456The monochromatic index of a connected graph. 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. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000543The size of the conjugacy class of a binary word. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000635The number of strictly order preserving maps of a poset into itself. St000690The size of the conjugacy class of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000732The number of double deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000762The sum of the positions of the weak records of an integer composition. St000823The number of unsplittable factors of the set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000844The size of the largest block in the direct sum decomposition of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. 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. St001118The acyclic chromatic index of a graph. 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. St001246The maximal difference between two consecutive entries of a permutation. St001344The neighbouring number of a permutation. St001346The number of parking functions that give the same permutation. St001388The number of non-attacking neighbors of a permutation. 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. St001490The number of connected components of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St001625The Möbius invariant of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001781The interlacing number of a set partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001838The number of nonempty primitive factors of a binary word. St001885The number of binary words with the same proper border set. St001889The size of the connectivity set of a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000105The number of blocks in the set partition. 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. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000247The number of singleton blocks 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. St000317The cycle descent number of a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000367The number of simsun double descents of a permutation. 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. St000462The major index minus the number of excedences of a permutation. 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. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes 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. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. St001545The second Elser number of a connected graph. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. 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. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. 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. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St000495The number of inversions of distance at most 2 of a permutation. St000831The number of indices that are either descents or recoils. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001875The number of simple modules with projective dimension at most 1. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. 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. St000717The number of ordinal summands of a poset. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000148The number of odd parts of a partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000228The size of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000517The Kreweras number of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000567The sum of the products of all pairs of parts. St000644The number of graphs with given frequency partition. St000674The number of hills of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. 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. St000913The number of ways to refine the partition into singletons. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000965The sum of the dimension of Ext^i(D(A),A) for i=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 $c_{n-1}$. St000992The alternating sum of the parts of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension 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. St001019Sum of the projective dimensions of the simple modules in 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. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001060The distinguishing index of a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001256Number of simple reflexive modules that are 2-stable reflexive. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001595The number of standard Young tableaux of the skew partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001910The height of the middle non-run of a Dyck path. St000102The charge of a semistandard tableau.
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