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Your data matches 919 different statistics following compositions of up to 3 maps.
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Matching statistic: St000329
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St001508
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
St001508: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
Description
The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.
Matching statistic: St000028
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St000053
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[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
Description
The number of valleys of the Dyck path.
Matching statistic: St000120
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[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
Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000155
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000306
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[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
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000331
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[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
Description
The number of upper interactions of a Dyck path.
An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St000651
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St000670
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000670: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
Description
The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
The following 909 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. 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. 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. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001530The depth of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000340The number of non-final maximal constant sub-paths of length greater than one. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000647The number of big descents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000783The side length of the largest staircase partition fitting into a partition. St000787The number of flips required to make a perfect matching noncrossing. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001115The number of even descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001394The genus of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001584The area statistic between a Dyck path and its bounce path. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000345The number of refinements of a partition. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000638The number of up-down runs of a permutation. St000808The number of up steps of the associated bargraph. St000912The number of maximal antichains in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 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. St001058The breadth of the ordered tree. St001093The detour number of a graph. St001187The number of simple modules with grade at least one 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001343The dimension of the reduced incidence algebra of a poset. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001674The number of vertices of the largest induced star graph in the graph. St001717The largest size of an interval in a poset. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000022The number of fixed points of a permutation. St000025The number of initial rises of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000035The number of left outer peaks of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000133The "bounce" of a permutation. St000156The Denert index of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000246The number of non-inversions of a permutation. St000272The treewidth of a graph. St000290The major index of a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. St000332The positive inversions of an alternating sign matrix. St000336The leg major index of a standard tableau. St000352The Elizalde-Pak rank of a permutation. St000354The number of recoils of a permutation. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000362The size of a minimal vertex cover of a graph. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000446The disorder of a permutation. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000538The number of even inversions of a permutation. 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. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. 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. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000648The number of 2-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000809The reduced reflection length of the permutation. 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. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000836The number of descents of distance 2 of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000961The shifted major index of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001011Number of simple modules of projective dimension 2 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. 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. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001114The number of odd descents of a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001270The bandwidth of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001427The number of descents of a signed permutation. St001485The modular major index of a binary word. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001665The number of pure excedances of a permutation. St001671Haglund's hag of 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. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001842The major index of a set partition. St001928The number of non-overlapping descents in a permutation. St001962The proper pathwidth of a graph. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000146The Andrews-Garvan crank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000288The number of ones in a binary word. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. 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. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000485The length of the longest cycle of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000619The number of cyclic descents of a permutation. St000628The balance of a binary word. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000767The number of runs in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000822The Hadwiger number of the graph. St000831The number of indices that are either descents or recoils. St000839The largest opener of a set partition. St000843The decomposition number of a perfect matching. St000883The number of longest increasing subsequences of a permutation. St000886The number of permutations with the same antidiagonal sums. St000925The number of topologically connected components of a set partition. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001029The size of the core of a graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001052The length of the exterior of a permutation. St001062The maximal size of a block of a set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001081The number of minimal length factorizations of a permutation into star transpositions. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001246The maximal difference between two consecutive entries of a permutation. St001280The number of parts of an integer partition that are at least two. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001471The magnitude of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001587Half of the largest even part of an integer partition. St001616The number of neutral elements in a lattice. St001670The connected partition number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001720The minimal length of a chain of small intervals in a lattice. St001733The number of weak left to right maxima of a Dyck path. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St000708The product of the parts of an integer partition. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000933The number of multipartitions of sizes given by an integer partition. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001279The sum of the parts of an integer partition that are at least two. St001486The number of corners of the ribbon associated with an integer composition. 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. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. 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 $c_{n-1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000216The absolute length of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001152The number of pairs with even minimum in a perfect matching. St000083The number of left oriented leafs of a binary tree except the first one. St000317The cycle descent number of a permutation. 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. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. 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. 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. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed 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). St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001346The number of parking functions that give the same permutation. St001555The order of a signed permutation. St000039The number of crossings of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000218The number of occurrences of the pattern 213 in a permutation. St000222The number of alignments in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000391The sum of the positions of the ones in a binary word. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000653The last descent of a permutation. St000663The number of right floats of a permutation. St000732The number of double deficiencies of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of 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. St000833The comajor index of a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001077The prefix exchange distance of a permutation. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001388The number of non-attacking neighbors of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001429The number of negative entries in a signed permutation. St001480The number of simple summands of the module J^2/J^3. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001592The maximal number of simple paths between any two different vertices of a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001721The degree of a binary word. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001896The number of right descents of a signed permutations. St000060The greater neighbor of the maximum. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. St000189The number of elements in the poset. St000455The second largest eigenvalue of a graph if it is integral. St000504The cardinality of the first block of a set partition. St000526The number of posets with combinatorially isomorphic order polytopes. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000717The number of ordinal summands of a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000823The number of unsplittable factors of the set partition. 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. St000906The length of the shortest maximal chain in a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001220The width of a permutation. 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. St001352The number of internal nodes in the modular decomposition of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001589The nesting number of a perfect matching. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001778The largest greatest common divisor of an element and its image in a permutation. St001884The number of borders of a binary word. St001959The product of the heights of the peaks of a Dyck path. St000104The number of facets in the order polytope of this poset. St000144The pyramid weight of the Dyck path. St000151The number of facets in the chain polytope of the poset. St000236The number of cyclical small weak excedances. St000643The size of the largest orbit of antichains under Panyushev complementation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001668The number of points of the poset minus the width of the poset. St001712The number of natural descents of a standard Young tableau. St001877Number of indecomposable injective modules with projective dimension 2. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000353The number of inner valleys of a permutation. St000761The number of ascents in an integer composition. St000872The number of very big descents of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001513The number of nested exceedences of a permutation. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001866The nesting alignments of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000092The number of outer peaks of a permutation. St000307The number of rowmotion orbits of a poset. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001863The number of weak excedances of a signed permutation. St001249Sum of the odd parts of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000023The number of inner peaks of a permutation. St000091The descent variation of a composition. St000148The number of odd parts of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000486The number of cycles of length at least 3 of a permutation. St000618The number of self-evacuating tableaux of given shape. St000779The tier of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001593This is the number of standard Young tableaux of the given shifted shape. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001871The number of triconnected components of a graph. St000099The number of valleys of a permutation, including the boundary. St000460The hook length of the last cell along the main diagonal of an integer partition. St000502The number of successions of a set partitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001151The number of blocks with odd minimum. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001820The size of the image of the pop stack sorting operator. St000824The sum of the number of descents and the number of recoils of a permutation. 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 $eA$. St000735The last entry on the main diagonal of a standard tableau. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St001100The 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 trees. 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. St001281The normalized isoperimetric number of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000420The number of Dyck paths that are weakly above a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000993The multiplicity of the largest part of an integer partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001808The box weight or horizontal decoration of a Dyck path. St000422The energy of a graph, if it is integral. St001626The number of maximal proper sublattices of a lattice. St000478Another weight of a partition according to Alladi. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St000456The monochromatic index of a connected graph. St000707The product of the factorials of the parts. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000667The greatest common divisor of the parts of the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a 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. 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. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. 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. St000944The 3-degree of an integer partition. 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 $eA$. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000026The position of the first return of a Dyck path. St000048The multinomial of the parts of a partition. St000075The orbit size of a standard tableau under promotion. St000182The number of permutations whose cycle type is the given integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000260The radius of a connected graph. St000296The length of the symmetric border of a binary word. St000393The number of strictly increasing runs in a binary word. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000699The toughness times the least common multiple of 1,. St000733The row containing the largest entry of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St000922The minimal number such that all substrings of this length are unique. St000928The sum of the coefficients of the character polynomial of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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 $g$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. 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. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001525The number of symmetric hooks on the diagonal of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001595The number of standard Young tableaux of the skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001614The cyclic permutation representation number of a skew partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001732The number of peaks visible from the left. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000117The number of centered tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000295The length of the border of a binary word. St000348The non-inversion sum of a binary word. St000369The dinv deficit of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000877The depth of the binary word interpreted as a path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001488The number of corners of a skew partition. St001541The Gini index of an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001814The number of partitions interlacing the given partition. St001910The height of the middle non-run of a Dyck path. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001930The weak major index of a binary word. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000806The semiperimeter of the associated bargraph. St000893The number of distinct diagonal sums of an alternating sign matrix. St001060The distinguishing index of a graph. 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. St001182Number of indecomposable injective modules with codominant dimension at least two 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. St000259The diameter of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000137The Grundy value of an integer partition. St000145The Dyson rank 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. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000327The number of cover relations in a poset. St000509The diagonal index (content) of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001175The size of a partition minus the hook length of the base cell. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001637The number of (upper) dissectors of a poset. St001763The Hurwitz number of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation 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. St000100The number of linear extensions of a poset. St000284The Plancherel distribution on integer partitions. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000909The number of maximal chains of maximal size in a poset. St000941The number of characters of the symmetric group whose value on the partition is even. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001545The second Elser number of a connected graph. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000219The number of occurrences of the pattern 231 in a permutation. St000477The weight of a partition according to Alladi. St000939The number of characters of the symmetric group whose value on the partition is positive. St001520The number of strict 3-descents. St000736The last entry in the first row of a semistandard tableau. St001569The maximal modular displacement of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000744The length of the path to the largest entry in a standard Young tableau. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001098The 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 vertex labelled trees. 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. St001118The acyclic chromatic index of a graph. St001130The number of two successive successions in a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. 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 $K[x]/(x^n)$. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001438The number of missing boxes of a skew partition. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000045The number of linear extensions of a binary tree. St000079The number of alternating sign matrices for a given Dyck path. St000174The flush statistic of a semistandard tableau. St000298The order dimension or Dushnik-Miller dimension of a poset. St000315The number of isolated vertices of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000958The number of Bruhat factorizations of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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 $K[x]/(x^n)$. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001487The number of inner corners of a skew partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000089The absolute variation of a composition. St000090The variation of a composition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000254The nesting number of a set partition. St000287The number of connected components of a graph. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000739The first entry in the last row of a semistandard tableau. St001566The length of the longest arithmetic progression in a permutation. St001623The number of doubly irreducible elements of a lattice. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000383The last part of an integer composition. St000519The largest length of a factor maximising the subword complexity. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000973The length of the boundary of an ordered tree. St000981The length of the longest zigzag subpath. St000230Sum of the minimal elements of the blocks of a set partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000464The Schultz index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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