Your data matches 373 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000644
(load all 3 compositions to match this statistic)
Mapping
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 0
[3]
 => 2
[2,1]
 => 2
[1,1,1]
 => 0
Description
The number of graphs with given frequency partition. The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur. For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$. There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.
Matching statistic: St001332
(load all 9 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
Description
The number of steps on the non-negative side of the walk associated with the permutation. Consider the walk taking an up step for each ascent, and a down step for each descent of the permutation. Then this statistic is the number of steps that begin and end at non-negative height.
Matching statistic: St001600
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Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001600: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => 1
[2]
 => [1,1]
 => [2]
 => 2
[1,1]
 => [2]
 => [1,1]
 => 0
[3]
 => [2,1]
 => [2,1]
 => 2
[2,1]
 => [2,1]
 => [2,1]
 => 2
[1,1,1]
 => [3]
 => [1,1,1]
 => 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs.
Matching statistic: St001785
(load all 2 compositions to match this statistic)
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001785: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => 1
[2]
 => [1,1]
 => [2]
 => 0
[1,1]
 => [2]
 => [1,1]
 => 2
[3]
 => [2,1]
 => [1,1,1]
 => 2
[2,1]
 => [2,1]
 => [1,1,1]
 => 2
[1,1,1]
 => [3]
 => [3]
 => 0
Description
The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. Given a partition $\lambda\vdash n$, let $\alpha(\lambda)$ be the partition given by the lengths of the antidiagonals of the Ferrers diagram of $\lambda$. Then, the value of the statistic on $\mu$ is the number of times $\mu$ appears in the multiset $\{\{\alpha(\lambda)\mid \lambda\vdash n\}\}$.
Matching statistic: St000155
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => [2,1,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,2,1] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,2,3,4] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,4,3,1] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,2,1] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,2,3,4,5] => 0
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000239
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000239: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
Description
The number of small weak excedances. A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$. This is the sum of [[St000022]] and [[St000237]].
Matching statistic: St000462
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [2,4,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,4,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [2,5,1,3,4] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,4,5,2] => 2
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000488
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Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000488: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => 2
Description
The number of cycles of a permutation of length at most 2.
Matching statistic: St000703
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => [2,1,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,2,1] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,2,3,4] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,4,3,1] => 2
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,2,1] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,2,3,4,5] => 0
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
Matching statistic: St000753
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000753: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 00 => 01 => 1
[2]
 => 100 => 010 => 000 => 0
[1,1]
 => 110 => 001 => 011 => 2
[3]
 => 1000 => 0110 => 0011 => 2
[2,1]
 => 1010 => 0000 => 0101 => 0
[1,1,1]
 => 1110 => 0011 => 0110 => 2
Description
The Grundy value for the game of Kayles on a binary word. Two players alternately may remove either a single 1 or two adjacent 1's. The player facing the word which has only 0's looses.
The following 363 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001119The length of a shortest maximal path in a graph. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001274The number of indecomposable injective modules with projective dimension equal to two. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001405The number of bonds in a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001729The number of visible descents of a permutation. St000026The position of the first return of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001439The number of even weak deficiencies and of odd weak exceedences. St001523The degree of symmetry of a Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St000673The number of non-fixed points of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000652The maximal difference between successive positions of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000830The total displacement of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000981The length of the longest zigzag subpath. St001498The normalised height of a Nakayama algebra with magnitude 1. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000741The Colin de Verdière graph invariant. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001279The sum of the parts of an integer partition that are at least two. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001669The number of single rises in a Dyck path. St000145The Dyson rank of a partition. St000260The radius of a connected graph. St000509The diagonal index (content) of a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St000015The number of peaks of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000759The smallest missing part in an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. 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 $c_{n-1}$. St000983The length of the longest alternating subword. St000995The largest even part of an integer partition. St001024Maximum of dominant dimensions of the simple modules 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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: 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 $eA$. St001239The largest vector space dimension of the double dual of a simple module 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001471The magnitude of a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001956The comajor index for set-valued two-row standard Young tableaux. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000027The major index 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. St000043The number of crossings plus two-nestings of a perfect matching. St000050The depth or height of a binary tree. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000084The number of subtrees. St000105The number of blocks in the set 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. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000134The size of the orbit of an alternating sign matrix under gyration. St000144The pyramid weight of the Dyck path. St000156The Denert index of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000240The number of indices that are not small excedances. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000305The inverse major index of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000340The number of non-final maximal constant sub-paths of length greater than one. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the 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. St000397The Strahler number of a rooted tree. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000461The rix statistic of a permutation. St000470The number of runs in a permutation. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000477The weight of a partition according to Alladi. St000484The sum of St000483 over all subsequences of length at least three. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000495The number of inversions of distance at most 2 of a permutation. St000501The size of the first part in the decomposition of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000518The number of distinct subsequences in a binary word. St000542The number of left-to-right-minima of a permutation. St000567The sum of the products of all pairs of parts. St000616The inversion index of a permutation. St000626The minimal period of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000638The number of up-down runs of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000682The Grundy value of Welter's game on a binary word. 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. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000733The row containing the largest entry 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. St000746The number of pairs with odd minimum in a perfect matching. 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. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000792The Grundy value for the game of ruler on a binary word. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. 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. St000828The spearman's rho of a permutation and the identity permutation. St000837The number of ascents of distance 2 of a permutation. St000839The largest opener of a set partition. St000842The breadth of a permutation. St000843The decomposition number of a perfect matching. St000844The size of the largest block in the direct sum decomposition of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000863The length of the first row of the shifted shape of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000886The number of permutations with the same antidiagonal sums. St000893The number of distinct diagonal sums of an alternating sign matrix. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000925The number of topologically connected components of a set partition. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000947The major index east count of a Dyck path. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000956The maximal displacement of a permutation. St000961The shifted major index of a permutation. St000979Half of MacMahon's equal index of a Dyck path. St000982The length of the longest constant subword. St000984The number of boxes below precisely one peak. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001034The area of the parallelogram polyomino associated with the Dyck path. St001042The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001094The depth index of a set 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. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. 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. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001285The number of primes in the column sums of the two line notation of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001313The number of Dyck paths above the lattice path given by a binary word. St001346The number of parking functions that give the same permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001379The number of inversions plus the major index of a permutation. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001485The modular major index of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001516The number of cyclic bonds of a permutation. St001524The degree of symmetry of a binary word. St001531Number of partial orders contained in the poset determined by the Dyck path. St001541The Gini index of an integer partition. St001545The second Elser number of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001566The length of the longest arithmetic progression in a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001657The number of twos in an integer partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001671Haglund's hag of a permutation. St001675The number of parts equal to the part in the reversed composition. St001696The natural major index of a standard Young tableau. St001699The major index of a standard tableau minus the weighted size of its shape. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001721The degree of a binary word. St001759The Rajchgot index of a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001808The box weight or horizontal decoration of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St001910The height of the middle non-run of a Dyck path. St001930The weak major index of a binary word. St001959The product of the heights of the peaks of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000259The diameter of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian.