Your data matches 609 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001133
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 4
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 4
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => 2
Description
The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001134
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001134: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 4
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 4
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => 2
Description
The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001809
(load all 5 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001809: 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]
 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 2
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000874
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000874: 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
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
Description
The position of the last double rise in a Dyck path. If the Dyck path has no double rises, this statistic is $0$.
Matching statistic: St000976
(load all 10 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000976: 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
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
Description
The sum of the positions of double up-steps of a Dyck path. This is part of MacMahon's equal index of a word, see [1, p. 135].
Matching statistic: St001800
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001800: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
Description
The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$. Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by projecting the Catalan path onto the $x,y$-plane and the $y,z$-plane, respectively. For a given Dyck path $D$ this is the number of Catalan paths $C$ such that $D_{xy}(C) = D_{yz}(C) = D$. If $D$ is of semilength $n$, $r_i(D)$ denotes the number of downsteps between the $i$-th and $(i+1)$-st upstep, and $s_i(D)$ denotes the number of upsteps between the $i$-th and $(i+1)$-st downstep, then this number is given by $\prod\limits_{i=1}^{n-1} \binom{r_i(D) + s_i(D)}{r_i(D)}$.
Matching statistic: St000290
(load all 3 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 00 => 0 = 2 - 2
[2]
 => 100 => 010 => 2 = 4 - 2
[1,1]
 => 110 => 001 => 0 = 2 - 2
[2,1]
 => 1010 => 0000 => 0 = 2 - 2
[2,2]
 => 1100 => 0101 => 2 = 4 - 2
[3,2,1]
 => 101010 => 000000 => 0 = 2 - 2
[4,3,2,1]
 => 10101010 => 00000000 => 0 = 2 - 2
[5,4,3,2,1]
 => 1010101010 => 0000000000 => 0 = 2 - 2
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000726
(load all 7 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 4 - 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0 = 2 - 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 2 - 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 4 - 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 2 - 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 2 - 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 0 = 2 - 2
Description
The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. The sum of the leaf labels is at least the size of the permutation, equality is attained for the binary trees that have only one leaf. This statistic is the sum of the leaf labels minus the size of the permutation.
Matching statistic: St001185
(load all 5 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 2 - 2
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 4 - 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0 = 2 - 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 2 - 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 4 - 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 2 - 2
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St001248
(load all 15 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => 0 = 2 - 2
[2]
 => [1,1]
 => [2]
 => 2 = 4 - 2
[1,1]
 => [2]
 => [1,1]
 => 0 = 2 - 2
[2,1]
 => [2,1]
 => [1,1,1]
 => 0 = 2 - 2
[2,2]
 => [2,2]
 => [2,1,1]
 => 2 = 4 - 2
[3,2,1]
 => [3,2,1]
 => [1,1,1,1,1,1]
 => 0 = 2 - 2
[4,3,2,1]
 => [4,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => 0 = 2 - 2
[5,4,3,2,1]
 => [5,4,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 0 = 2 - 2
Description
Sum of the even parts of a partition.
The following 599 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001279The sum of the parts of an integer partition that are at least two. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000054The first entry of the permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000402Half the size of the symmetry class of a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000439The position of the first down step of a Dyck path. St000505The biggest entry in the block containing the 1. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. 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. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement of a permutation. St000883The number of longest increasing subsequences of a permutation. St000961The shifted major index of 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 $c_{n-1}$. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001735The number of permutations with the same set of runs. St001959The product of the heights of the peaks of a Dyck path. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000012The area of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000055The inversion sum of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000255The number of reduced Kogan faces with the permutation as type. St000305The inverse major index of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000332The positive inversions of an alternating sign matrix. St000340The number of non-final maximal constant sub-paths of length greater than one. St000416The number of inequivalent increasing trees of an ordered tree. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000472The sum of the ascent bottoms of a permutation. St000501The size of the first part in the decomposition of a permutation. St000539The number of odd inversions of a permutation. St000542The number of left-to-right-minima of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000617The number of global maxima of a Dyck path. St000651The maximal size of a rise in a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000742The number of big ascents of a permutation after prepending zero. St000756The sum of the positions of the left to right maxima of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000984The number of boxes below precisely one peak. 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. St001077The prefix exchange distance of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001285The number of primes in the column sums of the two line notation of a permutation. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001375The pancake length of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001468The smallest fixpoint of a permutation. St001481The minimal height of a peak of a Dyck path. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001733The number of weak left to right maxima of a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001874Lusztig's a-function for the symmetric group. St000004The major index of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000133The "bounce" of a permutation. St000210Minimum over maximum difference of elements in cycles. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000235The number of indices that are not cyclical small weak excedances. St000237The number of small exceedances. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000293The number of inversions of a binary word. St000356The number of occurrences of the pattern 13-2. St000369The dinv deficit of a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000475The number of parts equal to 1 in a partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000616The inversion index of a permutation. St000653The last descent of a permutation. St000658The number of rises of length 2 of a Dyck path. St000673The number of non-fixed points of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000825The sum of the major and the inverse major index of a permutation. St000828The spearman's rho of a permutation and the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000995The largest even part of an integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001125The number of simple modules that satisfy the 2-regular condition 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001379The number of inversions plus the major index of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001402The number of separators in a permutation. St001510The number of self-evacuating linear extensions of a finite poset. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001671Haglund's hag of a permutation. St001696The natural major index of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001759The Rajchgot index of a permutation. St001814The number of partitions interlacing the given partition. St001902The number of potential covers of a poset. St000037The sign of a permutation. St000511The number of invariant subsets when acting with a permutation of given cycle type. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001485The modular major index of a binary word. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000316The number of non-left-to-right-maxima of a permutation. St000652The maximal difference between successive positions of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001388The number of non-attacking neighbors of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000222The number of alignments in the permutation. St000389The number of runs of ones of odd length in a binary word. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000471The sum of the ascent tops of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000873The aix statistic of a permutation. St000981The length of the longest zigzag subpath. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001411The number of patterns 321 or 3412 in a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001964The interval resolution global dimension of a poset. St001564The value of the forgotten symmetric functions when all variables set to 1. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001557The number of inversions of the second entry of a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000401The size of the symmetry class of a permutation. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000020The rank of the permutation. St000163The size of the orbit of the set partition under rotation. St000230Sum of the minimal elements of the blocks of a set partition. St000260The radius of a connected graph. St000420The number of Dyck paths that are weakly above a Dyck path. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000690The size of the conjugacy class of a permutation. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000738The first entry in the last row of a standard tableau. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. 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. 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 $K[x]/(x^n)$. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. 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. 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: St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001530The depth of a Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St000027The major index of a Dyck path. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000289The decimal representation of a binary word. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000391The sum of the positions of the ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000446The disorder of a permutation. St000461The rix statistic of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000792The Grundy value for the game of ruler on a binary word. St000833The comajor index of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001094The depth index of a set partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a 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 $A$. St001197The global 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001524The degree of symmetry of a binary word. St001669The number of single rises in a Dyck path. St001705The number of occurrences of the pattern 2413 in a permutation. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001058The breadth of the ordered tree. St001817The number of flag weak exceedances of a signed permutation. St000295The length of the border of a binary word. St000456The monochromatic index of a connected graph. St000529The number of permutations whose descent word is the given binary word. St000667The greatest common divisor of the parts of the partition. St000741The Colin de Verdière graph invariant. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000477The weight of a partition according to Alladi. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000478Another weight of a partition according to Alladi. St000630The length of the shortest palindromic decomposition of a binary word. St000694The number of affine bounded permutations that project to a given permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000990The first ascent of a permutation. St000997The even-odd crank of an integer partition. 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. St001471The magnitude of a Dyck path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000264The girth of a graph, which is not a tree. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000291The number of descents of a binary word. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000759The smallest missing part in an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. 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. St000847The number of standard Young tableaux whose descent set is the binary word. St000993The multiplicity of the largest part of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001732The number of peaks visible from the left. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001884The number of borders of a binary word. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000043The number of crossings plus two-nestings of a perfect matching. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. 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. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000296The length of the symmetric border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000481The number of upper covers of a partition in dominance order. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000629The defect of a binary word. St000664The number of right ropes of a permutation. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules 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. St001280The number of parts of an integer partition that are at least two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001541The Gini index of an integer partition. St001556The number of inversions of the third entry of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001910The height of the middle non-run of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000444The length of the maximal rise of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000137The Grundy value of an integer partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000284The Plancherel distribution on integer partitions. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000549The number of odd partial sums of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000628The balance of a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000913The number of ways to refine the partition into singletons. St000933The number of multipartitions of sizes given by an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. 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 $c_{n−1}$. St000982The length of the longest constant subword. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 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. 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001060The distinguishing index of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one 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. 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 $g$. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001372The length of a longest cyclic run of ones of a binary word. St001383The BG-rank of an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001722The number of minimal chains with small intervals between a binary word and the top element. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000488The number of cycles of a permutation of length at most 2. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000878The number of ones minus the number of zeros of a binary word. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001035The convexity degree of the parallelogram polyomino associated with 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001596The number of two-by-two squares inside a skew partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001697The shifted natural comajor index of a standard Young tableau. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001811The Castelnuovo-Mumford regularity of 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). St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition.