Your data matches 177 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000110
(load all 8 compositions to match this statistic)
Mapping
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 6
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001855
(load all 6 compositions to match this statistic)
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 3
[3,1,2] => [3,1,2] => 3
[3,2,1] => [3,2,1] => 6
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St000208
(load all 4 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,2] => ([(0,1)],2)
 => [1]
 => 1
[2,1] => ([],2)
 => [2]
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 2
[2,3,1] => ([(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([],3)
 => [3,3]
 => 6
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000228
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Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,2] => ([(0,1)],2)
 => [1]
 => 1
[2,1] => ([],2)
 => [2]
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 2
[2,3,1] => ([(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([],3)
 => [3,3]
 => 6
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000345
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,2] => ([(0,1)],2)
 => [1]
 => 1
[2,1] => ([],2)
 => [2]
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 2
[2,3,1] => ([(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([],3)
 => [3,3]
 => 6
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000479
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],1)
 => 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => ([],1)
 => 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 3
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 3
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 6
Description
The Ramsey number of a graph. This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
Matching statistic: St000531
(load all 4 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,2] => ([(0,1)],2)
 => [1]
 => 1
[2,1] => ([],2)
 => [2]
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 2
[2,3,1] => ([(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([],3)
 => [3,3]
 => 6
Description
The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St001389
(load all 4 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,2] => ([(0,1)],2)
 => [1]
 => 1
[2,1] => ([],2)
 => [2]
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 2
[2,3,1] => ([(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([],3)
 => [3,3]
 => 6
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001659
(load all 4 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,2] => ([(0,1)],2)
 => [1]
 => 1
[2,1] => ([],2)
 => [2]
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 2
[2,3,1] => ([(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([],3)
 => [3,3]
 => 6
Description
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
Matching statistic: St000020
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000020: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
 => [1] => [1] => 1
[1,2] => [[1,2]]
 => [1,2] => [1,2] => 1
[2,1] => [[1],[2]]
 => [2,1] => [2,1] => 2
[1,2,3] => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
[1,3,2] => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 2
[2,1,3] => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 3
[2,3,1] => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 2
[3,1,2] => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 3
[3,2,1] => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 6
Description
The rank of the permutation. This is its position among all permutations of the same size ordered lexicographically. This can be computed using the Lehmer code of a permutation: $$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$ where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.
The following 167 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000063The number of linear extensions of a certain poset defined for an integer partition. St000071The number of maximal chains in a poset. St000085The number of linear extensions of the tree. St000179The product of the hook lengths of the integer partition. St000780The size of the orbit under rotation of a perfect matching. St000909The number of maximal chains of maximal size in a poset. St000945The number of matchings in the dihedral orbit of a perfect matching. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001813The product of the sizes of the principal order filters in a poset. St000100The number of linear extensions of a poset. St001346The number of parking functions that give the same permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001808The box weight or horizontal decoration of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001645The pebbling number of a connected graph. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001624The breadth of a lattice. St000259The diameter of a connected graph. St001626The number of maximal proper sublattices of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St000014The number of parking functions supported by a Dyck path. St000144The pyramid weight of the Dyck path. St000184The size of the centralizer of any permutation of given cycle type. St000293The number of inversions of a binary word. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000477The weight of a partition according to Alladi. St000519The largest length of a factor maximising the subword complexity. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000784The maximum of the length and the largest part of the integer partition. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000922The minimal number such that all substrings of this length are unique. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. 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 $c_{n-1}$. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. 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. St001118The acyclic chromatic index of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001249Sum of the odd parts of a partition. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. 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. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001488The number of corners of a skew partition. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001733The number of weak left to right maxima of a Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001545The second Elser number of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000699The toughness times the least common multiple of 1,. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000717The number of ordinal summands of a poset. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000906The length of the shortest maximal chain in a poset. St001060The distinguishing index of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000736The last entry in the first row of a semistandard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000958The number of Bruhat factorizations of a permutation. St000997The even-odd crank of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St001959The product of the heights of the peaks of a Dyck path. St000075The orbit size of a standard tableau under promotion. St000177The number of free tiles in the pattern. St000178Number of free entries. St000302The determinant of the distance matrix of a connected graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation.