- FindStat

Your data matches 287 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000718
(load all 4 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]

Matching statistic: St000915
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Ore degree of a graph. This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.

Matching statistic: St001486
(load all 2 compositions to match this statistic)

Mapping

Mp00305: Permutations —parking function⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 3 = 2 + 1
[2,1] => [2,1] => [2,1] => 3 = 2 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 5 = 4 + 1
[1,3,2] => [1,3,2] => [1,3,2] => 5 = 4 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 4 = 3 + 1
[2,3,1] => [2,3,1] => [2,3,1] => 5 = 4 + 1
[3,1,2] => [3,1,2] => [3,1,2] => 4 = 3 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 5 = 4 + 1

Description

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

Matching statistic: St001643
(load all 3 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001643: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
 => 3 = 2 + 1
[2,1] => [1,2] => [1,0,1,0]
 => 3 = 2 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 5 = 4 + 1
[1,3,2] => [1,2,3] => [1,0,1,0,1,0]
 => 5 = 4 + 1
[2,1,3] => [1,2,3] => [1,0,1,0,1,0]
 => 5 = 4 + 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
 => 5 = 4 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 4 = 3 + 1
[3,2,1] => [1,3,2] => [1,0,1,1,0,0]
 => 4 = 3 + 1

Description

The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000301
(load all 6 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000301: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of facets of the stable set polytope of a graph. The stable set polytope of a graph

$G$ is the convex hull of the characteristic vectors of stable (or independent) sets of vertices of

$G$ inside

$\mathbb{R}^{V(G)}$ .

Matching statistic: St001213
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.

Matching statistic: St001259
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001259: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.

Matching statistic: St000235

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index

$i < n$ such that

$\pi_i = i+1$ , or the index

$i = n$ if

$\pi_n = 1$ .

Matching statistic: St000242

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index

$i$ such that

$\pi_i \in \{ i,i+1 \}$ considered cyclically.

Matching statistic: St000825

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000825: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 2
[2,1] => [1,2] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [2,3,1] => [1,3,2] => 4
[1,3,2] => [1,2,3] => [2,3,1] => [1,3,2] => 4
[2,1,3] => [1,2,3] => [2,3,1] => [1,3,2] => 4
[2,3,1] => [1,2,3] => [2,3,1] => [1,3,2] => 4
[3,1,2] => [1,3,2] => [2,1,3] => [3,1,2] => 3
[3,2,1] => [1,3,2] => [2,1,3] => [3,1,2] => 3

Description

The sum of the major and the inverse major index of a permutation. This statistic is the sum of [[St000004]] and [[St000305]].

The following 277 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001379The number of inversions plus the major index of a permutation. St001708The number of pairs of vertices of different degree in a graph. St001721The degree of a binary word. St001759The Rajchgot index of a permutation. St001861The number of Bruhat lower covers of a permutation. St000060The greater neighbor of the maximum. St000619The number of cyclic descents of a permutation. St000734The last entry in the first row of a standard tableau. St000841The largest opener of a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. 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. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001838The number of nonempty primitive factors of a binary word. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000395The sum of the heights of the peaks of a Dyck path. St000438The position of the last up step in a Dyck path. St000626The minimal period of a binary word. St000722The number of different neighbourhoods in a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001706The number of closed sets in a graph. St000108The number of partitions contained in the given partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000520The number of patterns in a permutation. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000656The number of cuts of a poset. St000690The size of the conjugacy class of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000189The number of elements in the poset. St000625The sum of the minimal distances to a greater element. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001717The largest size of an interval in a poset. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000289The decimal representation of a binary word. St000472The sum of the ascent bottoms of a permutation. St000493The los statistic of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000498The lcs statistic of a set partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000567The sum of the products of all pairs of parts. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000681The Grundy value of Chomp on Ferrers diagrams. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. 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. St001118The acyclic chromatic index of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001875The number of simple modules with projective dimension at most 1. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000354The number of recoils 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000829The Ulam distance of a permutation to the identity permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St000477The weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St000300The number of independent sets of vertices of a graph. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000770The major index of an integer partition when read from bottom to top. St000981The length of the longest zigzag subpath. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001834The number of non-isomorphic minors of a graph. St001959The product of the heights of the peaks of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000391The sum of the positions of the ones in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000490The intertwining number of a set partition. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000699The toughness times the least common multiple of 1,. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000792The Grundy value for the game of ruler on a binary word. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000844The size of the largest block in the direct sum decomposition of a permutation. St000849The number of 1/3-balanced pairs in a poset. St000867The sum of the hook lengths in the first row of an integer partition. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000914The sum of the values of the Möbius function of a poset. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St000990The first ascent of a permutation. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001077The prefix exchange distance of a permutation. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001281The normalized isoperimetric number of a graph. St001346The number of parking functions that give the same permutation. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001441The number of non-empty connected induced subgraphs of a graph. St001808The box weight or horizontal decoration of a Dyck path. St001815The number of order preserving surjections from a poset to a total order. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000624The normalized sum of the minimal distances to a greater element. St000653The last descent of a permutation. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). 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. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001372The length of a longest cyclic run of ones of a binary word. St001391The disjunction number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of 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. St001480The number of simple summands of the module J^2/J^3. St001592The maximal number of simple paths between any two different vertices of a graph. St001948The number of augmented double ascents of a permutation. St000422The energy of a graph, if it is integral. St001725The harmonious chromatic number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001060The distinguishing index of a graph. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001570The minimal number of edges to add to make a graph Hamiltonian. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000487The length of the shortest cycle 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000219The number of occurrences of the pattern 231 in a permutation. St000677The standardized bi-alternating inversion number of a permutation. St001114The number of odd descents of a permutation. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000634The number of endomorphisms of a poset. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000827The decimal representation of a binary word with a leading 1. St000045The number of linear extensions of a binary tree. St000100The number of linear extensions of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000418The number of Dyck paths that are weakly below a Dyck path. St000467The hyper-Wiener index of a connected graph. St000504The cardinality of the first block of a set partition. St000633The size of the automorphism group of a poset. St000678The number of up steps after the last double rise of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St000823The number of unsplittable factors of the set partition. St000910The number of maximal chains of minimal length in a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001637The number of (upper) dissectors of a poset. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000478Another weight of a partition according to Alladi. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. 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. St000659The number of rises of length at least 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000728The dimension of a set partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St000850The number of 1/2-balanced pairs in a poset. St000877The depth of the binary word interpreted as a path. St000919The number of maximal left branches of a binary tree. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St001371The length of the longest Yamanouchi prefix of a binary word. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000509The diagonal index (content) of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001917The order of toric promotion on the set of labellings of a graph. St000264The girth of a graph, which is not a tree. St001651The Frankl number of a lattice. St000302The determinant of the distance matrix of a connected graph. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. 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

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index 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. St001557The number of inversions of the second entry of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001703The villainy 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. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000311The number of vertices of odd degree in a graph. St000327The number of cover relations in a poset. St001345The Hamming dimension of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001754The number of tolerances of a finite lattice. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph.