Your data matches 132 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000137
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [2]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [3]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [4]
 => 0
Description
The Grundy value of an integer partition. Consider the two-player game on an integer partition. In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition. The first player that cannot move lose. This happens exactly when the empty partition is reached. The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1]. This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000142
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 0
Description
The number of even parts of a partition.
Matching statistic: St000148
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [2]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [3]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [4]
 => 0
Description
The number of odd parts of a partition.
Matching statistic: St000274
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000274: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
Description
The number of perfect matchings of a graph. A matching of a graph $G$ is a subset $F \subset E(G)$ such that no two edges in $F$ share a vertex in common. A perfect matching $F'$ is then a matching such that every vertex in $V(G)$ is incident with exactly one edge in $F'$.
Matching statistic: St000512
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [2]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [3]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [4]
 => 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000513
(load all 3 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000513: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 0
Description
The number of invariant subsets of size 2 when acting with a permutation of given cycle type.
Matching statistic: St000547
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 0
Description
The number of even non-empty partial sums of an integer partition.
Matching statistic: St000549
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000549: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [2]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [3]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [4]
 => 0
Description
The number of odd partial sums of an integer partition.
Matching statistic: St000620
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [2]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [3]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [4]
 => 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].
Matching statistic: St000811
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000811: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [2]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [3]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [4]
 => 0
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. For example, $p_{22} = s_{1111} - s_{211} + 2s_{22} - s_{31} + s_4$, so the statistic on the partition $22$ is 2. This is also the sum of the character values at the given conjugacy class over all irreducible characters of the symmetric group. [2] For a permutation $\pi$ of given cycle type, this is also the number of permutations whose square equals $\pi$. [2]
The following 122 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000928The sum of the coefficients of the character polynomial of an integer partition. St001071The beta invariant of the graph. St001092The number of distinct even parts of a partition. St001252Half the sum of the even parts of a partition. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001383The BG-rank of an integer partition. St001587Half of the largest even part of an integer partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001657The number of twos in an integer partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001783The number of odd automorphisms of a graph. St001827The number of two-component spanning forests of a graph. St000363The number of minimal vertex covers of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000911The number of maximal antichains of maximal size in a poset. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000309The number of vertices with even degree. St000313The number of degree 2 vertices of a graph. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000658The number of rises of length 2 of a Dyck path. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000877The depth of the binary word interpreted as a path. St000992The alternating sum of the parts of an integer partition. St001027Number of simple modules with projective dimension equal to injective 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. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001091The number of parts in an integer partition whose next smaller part has the same size. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001176The size of a partition minus its first part. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001371The length of the longest Yamanouchi prefix of a binary word. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001479The number of bridges of a graph. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001721The degree of a binary word. St001743The discrepancy of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001797The number of overfull subgraphs of a graph. St001826The maximal number of leaves on a vertex of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000048The multinomial of the parts of a partition. St000088The row sums of the character table of the symmetric group. St000160The multiplicity of the smallest part of a partition. St000315The number of isolated vertices of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000667The greatest common divisor of the parts of the partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. 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. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000917The open packing number of a graph. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001057The Grundy value of the game of creating an independent set in a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001616The number of neutral elements in a lattice. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001679The number of subsets of a lattice whose meet is the bottom element. St001691The number of kings in a graph. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000469The distinguishing number of a graph. St000636The hull number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001459The number of zero columns in the nullspace of a graph. St001619The number of non-isomorphic sublattices of a lattice. St001654The monophonic hull number of a graph. St001666The number of non-isomorphic subposets of a lattice which are lattices. 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. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph.