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Matching statistic: St000035
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St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000162
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St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000834
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St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 1 = 2 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 1 = 2 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 0 = 1 - 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000994
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(load all 77 compositions to match this statistic)
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St001269
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St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001665
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St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Matching statistic: St001729
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St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001737
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St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St001928
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St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000058
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Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => 2
[3,2,1] => [3,2,1] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
The following 793 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St001261The Castelnuovo-Mumford regularity of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001642The Prague dimension of a graph. St000028The number of stack-sorts needed to sort a permutation. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000884The number of isolated descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000010The length of the partition. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. 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. St000105The number of blocks in the set partition. 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. St000239The number of small weak excedances. St000258The burning number of a graph. St000259The diameter of a connected graph. St000299The number of nonisomorphic vertex-induced subtrees. 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. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000381The largest part of an integer composition. St000397The Strahler number of a rooted tree. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000638The number of up-down runs of a permutation. St000676The number of odd rises of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. 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. St000822The Hadwiger number of the graph. St000912The number of maximal antichains in a poset. St000918The 2-limited packing number of a graph. St000920The logarithmic height of a Dyck path. 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. St001029The size of the core of a graph. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. 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. St001315The dissociation number of a graph. 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. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. 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. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001512The minimum rank of a graph. St001555The order of a signed permutation. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001624The breadth of a lattice. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001717The largest size of an interval in a poset. St001720The minimal length of a chain of small intervals in a lattice. St001734The lettericity of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001746The coalition number of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001792The arboricity of a graph. St001814The number of partitions interlacing the given partition. St001863The number of weak excedances of a signed permutation. St001883The mutual visibility number of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001963The tree-depth of a graph. St000016The number of attacking pairs of a standard tableau. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. 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. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000256The number of parts from which one can substract 2 and still get an integer partition. St000260The radius of a connected graph. St000272The treewidth of a graph. St000291The number of descents 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000362The size of a minimal vertex cover of a graph. St000378The diagonal inversion number of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000628The balance of a binary word. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000783The side length of the largest staircase partition fitting into a partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001092The number of distinct even parts of a partition. St001153The number of blocks with even minimum in a set partition. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. 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. St001427The number of descents of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001484The number of singletons of an integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001726The number of visible inversions of a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001769The reflection length of a signed permutation. St001812The biclique partition number of a graph. St001864The number of excedances of a signed permutation. St001874Lusztig's a-function for the symmetric group. 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. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001935The number of ascents in a parking function. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. 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. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000071The number of maximal chains in a poset. St000079The number of alternating sign matrices for a given Dyck path. St000086The number of subgraphs. St000087The number of induced subgraphs. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000108The number of partitions contained in the given partition. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000120The number of left tunnels of a Dyck path. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000201The number of leaf nodes in a binary tree. St000244The cardinality of the automorphism group of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. 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. St000335The difference of lower and upper interactions. St000343The number of spanning subgraphs of a graph. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000382The first part of an integer composition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000396The register function (or Horton-Strahler number) of a binary tree. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000527The width of the poset. St000532The total number of rook placements on a Ferrers board. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000636The hull number of a graph. St000640The rank of the largest boolean interval in a poset. St000644The number of graphs with given frequency partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods 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. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000829The Ulam distance of a permutation to the identity permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000917The open packing number of a graph. 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. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000971The smallest closer of a set partition. St000972The composition number of a graph. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001057The Grundy value of the game of creating an independent set in a graph. St001058The breadth of the ordered tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001345The Hamming dimension of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001480The number of simple summands of the module J^2/J^3. St001488The number of corners of a skew partition. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St001589The nesting number of a perfect matching. St001597The Frobenius rank of a skew partition. St001616The number of neutral elements in a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001679The number of subsets of a lattice whose meet is the bottom element. 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. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St001754The number of tolerances of a finite lattice. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001802The number of endomorphisms of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001820The size of the image of the pop stack sorting operator. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001884The number of borders of a binary word. St001896The number of right descents of a signed permutations. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000053The number of valleys of the Dyck path. St000055The inversion sum of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000081The number of edges of a graph. St000094The depth of an ordered tree. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000133The "bounce" of a permutation. St000143The largest repeated part of a partition. St000144The pyramid weight of the Dyck path. St000148The number of odd parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000157The number of descents of a standard tableau. St000168The number of internal nodes of an ordered tree. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000228The size of a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000290The major index of a binary word. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000330The (standard) major index of a standard tableau. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000336The leg major index of a standard tableau. St000353The number of inner valleys of a permutation. St000361The second Zagreb index of a graph. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000386The number of factors DDU in a Dyck path. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000439The position of the first down step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000461The rix statistic of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000475The number of parts equal to 1 in a partition. St000486The number of cycles of length at least 3 of a permutation. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000538The number of even inversions of a permutation. St000546The number of global descents of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000632The jump number of the poset. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000778The metric dimension of a graph. St000779The tier of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000797The stat`` of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000836The number of descents of distance 2 of 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. St000869The sum of the hook lengths of an integer partition. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000891The number of distinct diagonal sums of a permutation matrix. St000932The number of occurrences of the pattern UDU in a Dyck path. 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 $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in 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. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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). St001085The number of occurrences of the vincular pattern |21-3 in a permutation. 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. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. 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. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension 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. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001341The number of edges in the center of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001375The pancake length of a permutation. St001391The disjunction number of a graph. St001394The genus of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length 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. St001424The number of distinct squares in a binary word. St001429The number of negative entries in a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001469The holeyness of a permutation. St001479The number of bridges of a graph. St001485The modular major index of a binary word. 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. 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. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. 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. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices 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. St001649The length of a longest trail in a graph. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001671Haglund's hag of a permutation. St001684The reduced word complexity of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001777The number of weak descents in an integer composition. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001823The Stasinski-Voll length of a signed permutation. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001895The oddness of a signed permutation. St001930The weak major index of a binary word. St001931The weak major index of an integer composition regarded as a word. St001946The number of descents in a parking function. St001949The rigidity index of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. 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. 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. 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. St001569The maximal modular displacement of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000677The standardized bi-alternating inversion number of a permutation. St000809The reduced reflection length of the permutation. St000919The number of maximal left branches of a binary tree. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000925The number of topologically connected components of a set partition. St000933The number of multipartitions of sizes given by an integer partition. St001062The maximal size of a block of a set partition. St000083The number of left oriented leafs of a binary tree except the first one. St000389The number of runs of ones of odd length in a binary word. St000472The sum of the ascent bottoms 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. St000558The number of occurrences of the pattern {{1,2}} in a set partition. 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. St000658The number of rises of length 2 of a Dyck path. 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. St001114The number of odd descents of a permutation. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001668The number of points of the poset minus the width of the poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000307The number of rowmotion orbits of a poset. St000418The number of Dyck paths that are weakly below a Dyck path. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000823The number of unsplittable factors of the set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. 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. St000990The first ascent of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001346The number of parking functions that give the same permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001959The product of the heights of the peaks of a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. 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. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000874The position of the last double rise in a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions 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. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001077The prefix exchange distance of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001626The number of maximal proper sublattices of a lattice. St001948The number of augmented double ascents of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000642The size of the smallest orbit of antichains under Panyushev complementation. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000420The number of Dyck paths that are weakly above a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000464The Schultz index of a connected graph. St000487The length of the shortest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000674The number of hills of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. 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. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000993The multiplicity of the largest part of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001249Sum of the odd parts of a partition. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001360The number of covering relations in Young's lattice below a partition. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St001545The second Elser number of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001808The box weight or horizontal decoration of a Dyck path. St001885The number of binary words with the same proper border set. St001933The largest multiplicity of a part in an integer partition. St000219The number of occurrences of the pattern 231 in a permutation. St000653The last descent of a permutation. St000794The mak of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000667The greatest common divisor of the parts of the partition. 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. St000997The even-odd crank of an integer partition. St001060The distinguishing index of a graph. 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. St001118The acyclic chromatic index of a graph. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension 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 $K[x]/(x^n)$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000177The number of free tiles in the pattern. St000178Number of free entries. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St001811The Castelnuovo-Mumford regularity of a permutation.
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