Your data matches 264 different statistics following compositions of up to 3 maps.
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St001074: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => 2 = 1 + 1
[2,1] => 2 = 1 + 1
[1,2,3] => 3 = 2 + 1
[1,3,2] => 5 = 4 + 1
[2,1,3] => 3 = 2 + 1
[2,3,1] => 5 = 4 + 1
[3,1,2] => 3 = 2 + 1
[3,2,1] => 3 = 2 + 1
Description
The number of inversions of the cyclic embedding of a permutation. The cyclic embedding of a permutation $\pi$ of length $n$ is given by the permutation of length $n+1$ represented in cycle notation by $(\pi_1,\ldots,\pi_n,n+1)$. This reflects in particular the fact that the number of long cycles of length $n+1$ equals $n!$. This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length $n$ equals $n!\cdot(3n-1)/12$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
St000033: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [2,1,3] => 4
[3,2,1] => [1,3,2] => [2,1,3] => 4
Description
The number of permutations greater than or equal to the given permutation in (strong) Bruhat order.
Mp00114: Permutations connectivity setBinary words
Mp00262: Binary words poset of factorsPosets
St000327: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => ([(0,1)],2)
=> 1
[2,1] => 0 => ([(0,1)],2)
=> 1
[1,2,3] => 11 => ([(0,2),(2,1)],3)
=> 2
[1,3,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1] => 00 => ([(0,2),(2,1)],3)
=> 2
[3,1,2] => 00 => ([(0,2),(2,1)],3)
=> 2
[3,2,1] => 00 => ([(0,2),(2,1)],3)
=> 2
Description
The number of cover relations in a poset. Equivalently, this is also the number of edges in the Hasse diagram [1].
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
St000545: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [2,1,3] => 4
[3,2,1] => [1,3,2] => [2,1,3] => 4
Description
The number of parabolic double cosets with minimal element being the given permutation. For $w \in S_n$, this is $$\big| W_I \tau W_J\ :\ \tau \in S_n,\ I,J \subseteq S,\ w = \min\{W_I \tau W_J\}\big|$$ where $S$ is the set of simple transpositions, $W_K$ is the parabolic subgroup generated by $K \subseteq S$, and $\min\{W_I \tau W_J\}$ is the unique minimal element in weak order in the double coset $W_I \tau W_J$. [1] contains a combinatorial description of these parabolic double cosets which can be used to compute this statistic.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St000958: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [3,2,1] => 4
[3,2,1] => [1,3,2] => [3,2,1] => 4
Description
The number of Bruhat factorizations of a permutation. This is the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that the number of inversions of $t_1 \cdots t_i$ equals $i$ for all $1 \leq i \leq \ell$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00310: Permutations toric promotionPermutations
St001468: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,2,3] => [3,2,1] => 2
[2,1,3] => [1,2,3] => [3,2,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 2
[3,1,2] => [1,3,2] => [2,3,1] => 4
[3,2,1] => [1,3,2] => [2,3,1] => 4
Description
The smallest fixpoint of a permutation. A fixpoint of a permutation of length $n$ if an index $i$ such that $\pi(i) = i$, and we set $\pi(n+1) = n+1$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St001684: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [3,2,1] => 4
[3,2,1] => [1,3,2] => [3,2,1] => 4
Description
The reduced word complexity of a permutation. For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$. For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$. This statistic appears in [1, Question 6.1].
Mp00065: Permutations permutation posetPosets
Mp00074: Posets to graphGraphs
St001917: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> ([],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 4
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 4
[3,2,1] => ([],3)
=> ([],3)
=> 2
Description
The order of toric promotion on the set of labellings of a graph. In the context of toric promotion, a labelling of a graph $(V, E)$ with $n=|V|$ vertices is a bijection $\sigma: V \to [n]$. In particular, any graph has $n!$ labellings.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St001938: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [3]
=> 2
[1,3,2] => [1,2,3] => [3]
=> 2
[2,1,3] => [1,2,3] => [3]
=> 2
[2,3,1] => [1,2,3] => [3]
=> 2
[3,1,2] => [1,3,2] => [2,1]
=> 4
[3,2,1] => [1,3,2] => [2,1]
=> 4
Description
The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. Let $\pi$ be a permutation of cycle type $\mu$. A transitive monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ is a tuple of $r = n + \ell(\mu) - 2$ transpositions $$ (a_1, b_1),\dots,(a_r, b_r) $$ with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, such that the subgroup of $\mathfrak S_n$ generated by the transpositions acts transitively on $\{1,\dots,n\}$ and hose product, in this order, is $\pi$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St000111: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 2 = 1 + 1
[2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[1,3,2] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[2,1,3] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[2,3,1] => [1,2,3] => [2,3,1] => 3 = 2 + 1
[3,1,2] => [1,3,2] => [3,2,1] => 5 = 4 + 1
[3,2,1] => [1,3,2] => [3,2,1] => 5 = 4 + 1
Description
The sum of the descent tops (or Genocchi descents) of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_i.$$
The following 254 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000341The non-inversion sum of a permutation. St000530The number of permutations with the same descent word as the given permutation. St001127The sum of the squares of the parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001671Haglund's hag of a permutation. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001846The number of elements which do not have a complement in the lattice. St000063The number of linear extensions of a certain poset defined for an integer partition. 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. St000912The number of maximal antichains in a poset. St001400The total number of Littlewood-Richardson tableaux of given shape. St001782The order of rowmotion on the set of order ideals of a poset. St001834The number of non-isomorphic minors of a graph. St001930The weak major index of a binary word. St000012The area of a Dyck path. St000020The rank of the permutation. St000040The number of regions of the inversion arrangement of a permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000288The number of ones in a binary word. St000402Half the size of the symmetry class of a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000984The number of boxes below precisely one peak. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^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. St001592The maximal number of simple paths between any two different vertices of a graph. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001651The Frankl number of a lattice. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St001806The upper middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001813The product of the sizes of the principal order filters in a poset. St001865The number of alignments of a signed permutation. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000022The number of fixed points of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000055The inversion sum of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000088The row sums of the character table of the symmetric group. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000148The number of odd parts of a partition. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000185The weighted size of a partition. St000215The number of adjacencies of a permutation, zero appended. St000221The number of strong fixed points of a permutation. St000224The sorting index of a permutation. St000231Sum of the maximal elements of the blocks of a set partition. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000248The number of anti-singletons of a set partition. St000277The number of ribbon shaped standard tableaux. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000289The decimal representation of a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. St000309The number of vertices with even degree. St000313The number of degree 2 vertices of a graph. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000338The number of pixed points of a permutation. St000391The sum of the positions of the ones in a binary word. St000438The position of the last up step in a Dyck path. St000446The disorder of a permutation. St000461The rix statistic of a permutation. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000475The number of parts equal to 1 in a partition. St000494The number of inversions of distance at most 3 of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. 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. St000656The number of cuts of a poset. St000677The standardized bi-alternating inversion number of a permutation. St000680The Grundy value for Hackendot on posets. St000692Babson and Steingrímsson's statistic of a permutation. St000721The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching. St000753The Grundy value for the game of Kayles on a binary word. 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. St000827The decimal representation of a binary word with a leading 1. St000833The comajor index of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000868The aid statistic in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000946The sum of the skew hook positions in a Dyck path. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001077The prefix exchange distance of a permutation. St001094The depth index of a set partition. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001312Number of parabolic noncrossing partitions indexed by the composition. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001375The pancake length of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. 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. 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001692The number of vertices with higher degree than the average degree in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001708The number of pairs of vertices of different degree in a graph. St001721The degree of a binary word. St001736The total number of cycles in a graph. St001799The number of proper separations of a graph. St001874Lusztig's a-function for the symmetric group. St001910The height of the middle non-run of a Dyck path. St001926Sparre Andersen's position of the maximum of a signed permutation. St001956The comajor index for set-valued two-row standard Young tableaux. St001961The sum of the greatest common divisors of all pairs of parts. St000014The number of parking functions supported by a Dyck path. St000145The Dyson rank of a partition. St000294The number of distinct factors of a binary word. St000351The determinant of the adjacency matrix of a graph. 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. 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. St000756The sum of the positions of the left to right maxima of a permutation. St000825The sum of the major and the inverse major index of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. 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. 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. St000997The even-odd crank of an integer partition. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001706The number of closed sets in a graph. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000454The largest eigenvalue of a graph if it is integral. 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. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. 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. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000981The length of the longest zigzag subpath. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001378The product of the cohook lengths of the integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001644The dimension of a graph. St001645The pebbling number of a connected graph. St001812The biclique partition number of a graph. St000095The number of triangles of a graph. St000260The radius of a connected graph. St000299The number of nonisomorphic vertex-induced subtrees. St000450The number of edges minus the number of vertices plus 2 of a 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. St000822The Hadwiger number of the graph. 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 $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001742The difference of the maximal and the minimal degree in a graph. 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. St000553The number of blocks of a graph. St000674The number of hills of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. 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. St001391The disjunction number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000509The diagonal index (content) of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000464The Schultz index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. 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. 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. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001330The hat guessing number of a graph. 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. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000937The number of positive values of the symmetric group character corresponding to the partition.