Your data matches 568 different statistics following compositions of up to 3 maps.
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Matching statistic: St001935
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Mapping
St001935: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 1
[2,1] => 0
[1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[2,1,1] => 0
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 0
[1,2,2] => 1
[2,1,2] => 1
[2,2,1] => 0
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
Description
The number of ascents in a parking function. This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St001946
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Mapping
St001946: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 0
[2,1] => 1
[1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 1
[2,1,1] => 1
[1,1,3] => 0
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 0
[2,1,2] => 1
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
Description
The number of descents in a parking function. This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St000021
(load all 20 compositions to match this statistic)
Mapping
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,2] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => 0
[1,1,2] => [1,2,3] => 0
[1,2,1] => [1,3,2] => 1
[2,1,1] => [2,3,1] => 1
[1,1,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => 1
[3,1,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => 0
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 2
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000662
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Mapping
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,2] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => 0
[1,1,2] => [1,2,3] => 0
[1,2,1] => [1,3,2] => 1
[2,1,1] => [2,3,1] => 1
[1,1,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => 1
[3,1,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => 0
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 2
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000761
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Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,1,1] => [1,1,1] => 0
[1,1,2] => [1,1,2] => 1
[1,2,1] => [1,2,1] => 1
[2,1,1] => [2,1,1] => 0
[1,1,3] => [1,1,3] => 1
[1,3,1] => [1,3,1] => 1
[3,1,1] => [3,1,1] => 0
[1,2,2] => [1,2,2] => 1
[2,1,2] => [2,1,2] => 1
[2,2,1] => [2,2,1] => 0
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 0
Description
The number of ascents in an integer composition. A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
Matching statistic: St001489
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Mapping
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,2] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => 0
[1,1,2] => [1,2,3] => 0
[1,2,1] => [1,3,2] => 1
[2,1,1] => [2,3,1] => 1
[1,1,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => 1
[3,1,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => 0
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 2
Description
The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St001761
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Mapping
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,2] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => 0
[1,1,2] => [1,2,3] => 0
[1,2,1] => [1,3,2] => 1
[2,1,1] => [2,3,1] => 1
[1,1,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => 1
[3,1,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => 0
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 2
Description
The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000325
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Mapping
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [2,3,1] => 2 = 1 + 1
[1,1,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => 2 = 1 + 1
[1,2,2] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [3,1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => 2 = 1 + 1
[3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 3 = 2 + 1
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
Matching statistic: St000381
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Mapping
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,1] => [1,1] => 1 = 0 + 1
[1,2] => [1,1] => 1 = 0 + 1
[2,1] => [2] => 2 = 1 + 1
[1,1,1] => [1,1,1] => 1 = 0 + 1
[1,1,2] => [1,1,1] => 1 = 0 + 1
[1,2,1] => [1,2] => 2 = 1 + 1
[2,1,1] => [2,1] => 2 = 1 + 1
[1,1,3] => [1,2] => 2 = 1 + 1
[1,3,1] => [1,1,1] => 1 = 0 + 1
[3,1,1] => [2,1] => 2 = 1 + 1
[1,2,2] => [1,1,1] => 1 = 0 + 1
[2,1,2] => [2,1] => 2 = 1 + 1
[2,2,1] => [1,2] => 2 = 1 + 1
[1,2,3] => [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,2] => 2 = 1 + 1
[2,1,3] => [2,1] => 2 = 1 + 1
[2,3,1] => [1,2] => 2 = 1 + 1
[3,1,2] => [2,1] => 2 = 1 + 1
[3,2,1] => [3] => 3 = 2 + 1
Description
The largest part of an integer composition.
Matching statistic: St000470
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Mapping
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [2,3,1] => 2 = 1 + 1
[1,1,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => 2 = 1 + 1
[1,2,2] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [3,1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => 2 = 1 + 1
[3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 3 = 2 + 1
Description
The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.
The following 558 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000808The number of up steps of the associated bargraph. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001427The number of descents of a signed permutation. St001644The dimension of a graph. St001726The number of visible inversions of a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001823The Stasinski-Voll length of a signed permutation. St001896The number of right descents of a signed permutations. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000010The length of the partition. 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. St000071The number of maximal chains in a poset. 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. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000482The (zero)-forcing number of a graph. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000883The number of longest increasing subsequences of a permutation. St000935The number of ordered refinements of an integer partition. 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. St001029The size of the core of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. 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. 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. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. 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. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000067The inversion number of the alternating sign matrix. St000080The rank of the poset. St000091The descent variation of a composition. St000120The number of left tunnels of a Dyck path. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000224The sorting index of a permutation. St000228The size of a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000288The number of ones in a binary word. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000441The number of successions of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000670The reversal length of a permutation. St000691The number of changes of a binary word. St000731The number of double exceedences of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000753The Grundy value for the game of Kayles on a binary word. St000778The metric dimension of a graph. St000784The maximum of the length and the largest part of the integer partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000868The aid statistic in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001127The sum of the squares of the parts of a partition. 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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001375The pancake length of a permutation. St001391The disjunction number of a graph. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001684The reduced word complexity of a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001869The maximum cut size of a graph. St001905The number of preferred parking spots in a parking function less than the index of the car. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001948The number of augmented double ascents of a permutation. St001949The rigidity index of a graph. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000063The number of linear extensions of a certain poset defined for an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000087The number of induced subgraphs. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000532The total number of rook placements on a Ferrers board. St000628The balance of a binary word. St000636The hull number of a graph. 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. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000909The number of maximal chains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000942The number of critical left to right maxima of the parking functions. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001050The number of terminal closers of a set partition. St001058The breadth of the ordered tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001399The distinguishing number of a poset. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001497The position of the largest weak excedence of a permutation. St001530The depth of a Dyck path. St001589The nesting number of a perfect matching. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number 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. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000094The depth of an ordered tree. St000299The number of nonisomorphic vertex-induced subtrees. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. 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 $c_{n-1}$. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. 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. St001674The number of vertices of the largest induced star graph in the graph. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000354The number of recoils of a permutation. St000539The number of odd inversions of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000730The maximal arc length of a set partition. St001061The number of indices that are both descents and recoils of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. 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. St001480The number of simple summands of the module J^2/J^3. St001592The maximal number of simple paths between any two different vertices of a graph. St000444The length of the maximal rise of a Dyck path. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000297The number of leading ones in a binary word. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. 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. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001418Half of the global dimension of the stable Auslander algebra of 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. St001668The number of points of the poset minus the width of the poset. St000061The number of nodes on the left branch of a binary tree. St000485The length of the longest cycle of a permutation. 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. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000910The number of maximal chains of minimal length in a poset. St000925The number of topologically connected components of a set partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001624The breadth of a lattice. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. 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. St001280The number of parts of an integer partition that are at least two. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000455The second largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001487The number of inner corners of a skew partition. St000137The Grundy value of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000699The toughness times the least common multiple of 1,. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St000466The Gutman (or modified Schultz) index of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000177The number of free tiles in the pattern. St000178Number of free entries. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000460The hook length of the last cell along the main diagonal of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 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. St000693The modular (standard) major index of a standard tableau. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000934The 2-degree of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001651The Frankl number of a lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. 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. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000478Another weight of a partition according to Alladi. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001943The sum of the squares of the hook lengths of an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. 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. 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. St000075The orbit size of a standard tableau under promotion.