- FindStat

Your data matches 185 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000063
(load all 9 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 2
[1,2] => [1,2] => [2]
 => 3
[2,1] => [1,2] => [2]
 => 3
[1,2,3] => [1,2,3] => [3]
 => 4
[1,3,2] => [1,2,3] => [3]
 => 4
[2,1,3] => [1,2,3] => [3]
 => 4
[2,3,1] => [1,2,3] => [3]
 => 4
[3,1,2] => [1,3,2] => [2,1]
 => 6
[3,2,1] => [1,3,2] => [2,1]
 => 6

Description

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition

$\lambda$ also counts cover-inclusive Dyck tilings of

$\lambda\setminus\mu$ , summed over all

$\mu$ , as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

Matching statistic: St000070
(load all 3 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 2
[1,2] => [1,2] => ([(0,1)],2)
 => 3
[2,1] => [2,1] => ([(0,1)],2)
 => 3
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4

Description

The number of antichains in a poset. An antichain in a poset

$P$ is a subset of elements of

$P$ which are pairwise incomparable. An order ideal is a subset

$I$ of

$P$ such that

$a\in I$ and

$b \leq_P a$ implies

$b \in I$ . Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.

Matching statistic: St000104
(load all 2 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000104: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 2
[1,2] => [1,2] => ([(0,1)],2)
 => 3
[2,1] => [2,1] => ([(0,1)],2)
 => 3
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4

Description

The number of facets in the order polytope of this poset.

Matching statistic: St000151
(load all 2 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000151: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 2
[1,2] => [1,2] => ([(0,1)],2)
 => 3
[2,1] => [2,1] => ([(0,1)],2)
 => 3
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 4

Description

The number of facets in the chain polytope of the poset.

Matching statistic: St000912
(load all 2 compositions to match this statistic)

Mapping

Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 6
[1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 6

Description

The number of maximal antichains in a poset.

Matching statistic: St001400
(load all 9 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001400: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 2
[1,2] => [1,2] => [2]
 => 3
[2,1] => [1,2] => [2]
 => 3
[1,2,3] => [1,2,3] => [3]
 => 4
[1,3,2] => [1,2,3] => [3]
 => 4
[2,1,3] => [1,2,3] => [3]
 => 4
[2,3,1] => [1,2,3] => [3]
 => 4
[3,1,2] => [1,3,2] => [2,1]
 => 6
[3,2,1] => [1,3,2] => [2,1]
 => 6

Description

The total number of Littlewood-Richardson tableaux of given shape. This is the multiplicity of the Schur function

$s_\lambda$ in

$\sum_{\mu, \nu} s_\mu s_\nu$ , where the sum is over all partitions

$\mu$ and

$\nu$ .

Matching statistic: St001782
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001782: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 2
[1,2] => [1,2] => ([(0,1)],2)
 => 3
[2,1] => [1,2] => ([(0,1)],2)
 => 3
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => 6
[3,2,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 6

Description

The order of rowmotion on the set of order ideals of a poset.

Matching statistic: St001834

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001834: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 2
[1,2] => [1,2] => ([],2)
 => 3
[2,1] => [1,2] => ([],2)
 => 3
[1,2,3] => [1,2,3] => ([],3)
 => 4
[1,3,2] => [1,2,3] => ([],3)
 => 4
[2,1,3] => [1,2,3] => ([],3)
 => 4
[2,3,1] => [1,2,3] => ([],3)
 => 4
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => 6
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => 6

Description

The number of non-isomorphic minors of a graph. A minor of a graph

$G$ is a graph obtained from

$G$ by repeatedly deleting or contracting edges, or removing isolated vertices. This statistic records the total number of (non-empty) non-isomorphic minors of a graph.

Matching statistic: St000530
(load all 11 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000530: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [2,1] => 1 = 2 - 1
[1,2] => [1,0,1,0]
 => [3,1,2] => 2 = 3 - 1
[2,1] => [1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 6 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 6 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [4,3,1,2] => 3 = 4 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1

Description

The number of permutations with the same descent word as the given permutation. The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation. This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.

Matching statistic: St001127
(load all 3 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001127: 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] => [1,2] => [1,1]
 => 2 = 3 - 1
[2,1] => [1,2] => [1,1]
 => 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[1,3,2] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[2,1,3] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[2,3,1] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[3,1,2] => [1,3,2] => [2,1]
 => 5 = 6 - 1
[3,2,1] => [1,3,2] => [2,1]
 => 5 = 6 - 1

Description

The sum of the squares of the parts of a partition.

The following 175 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001360The number of covering relations in Young's lattice below a partition. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001684The reduced word complexity of a permutation. St000014The number of parking functions supported by a Dyck path. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000294The number of distinct factors of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000468The Hosoya index of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000825The sum of the major and the inverse major index of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001616The number of neutral elements in a lattice. St001779The order of promotion on the set of linear extensions of a poset. St001808The box weight or horizontal decoration of a Dyck path. St000018The number of inversions of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000231Sum of the maximal elements of the blocks of a set partition. St000277The number of ribbon shaped standard tableaux. St000391The sum of the positions of the ones in a binary word. St000494The number of inversions of distance at most 3 of a permutation. St000529The number of permutations whose descent word is the given binary word. St000721The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching. St000795The mad of a permutation. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001671Haglund's hag of a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001592The maximal number of simple paths between any two different vertices of a graph. St001651The Frankl number of a lattice. St001846The number of elements which do not have a complement in the lattice. St000327The number of cover relations in a poset. St000569The sum of the heights of the vertices of a binary tree. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000438The position of the last up step in a Dyck path. St000471The sum of the ascent tops of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000890The number of nonzero entries in an alternating sign matrix. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001561The value of the elementary symmetric function evaluated at 1. St001802The number of endomorphisms of a graph. St001865The number of alignments of a signed permutation. St000248The number of anti-singletons of a set partition. St000289The decimal representation of a binary word. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. 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. St000680The Grundy value for Hackendot on posets. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000946The sum of the skew hook positions in a Dyck path. St001077The prefix exchange distance of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001721The degree of a binary word. St001926Sparre Andersen's position of the maximum of a signed permutation. St000878The number of ones minus the number of zeros of a binary word. St000997The even-odd crank of an integer partition. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St000454The largest eigenvalue of a graph if it is integral. St000553The number of blocks of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001391The disjunction number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000450The number of edges minus the number of vertices plus 2 of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St001812The biclique partition number of a graph. 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. St000762The sum of the positions of the weak records of an integer composition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000673The number of non-fixed points of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement of a permutation. 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. 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

$eA$ . St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000981The length of the longest zigzag subpath. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000227The osculating paths major index of an alternating sign matrix. St000247The number of singleton blocks of a set partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000674The number of hills of a Dyck path. St000894The trace of an alternating sign matrix. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. 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. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000509The diagonal index (content) of a partition. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000699The toughness times the least common multiple of 1,. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St000101The cocharge of a semistandard tableau. St001117The game chromatic index of a graph. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001649The length of a longest trail in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001820The size of the image of the pop stack sorting operator. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000736The last entry in the first row of a semistandard tableau. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001569The maximal modular displacement of a permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001720The minimal length of a chain of small intervals in a lattice. St001742The difference of the maximal and the minimal degree in a graph. St000102The charge of a semistandard tableau. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001556The number of inversions of the third entry of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001856The number of edges in the reduced word graph of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset.