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Your data matches 92 different statistics following compositions of up to 3 maps.
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Matching statistic: St001622
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Values
([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000189
Values
([(0,1)],2)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> 3
Description
The number of elements in the poset.
Matching statistic: St001636
Values
([(0,1)],2)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> 3
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
Matching statistic: St000228
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Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([],1)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> [2,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000459
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Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([],1)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> [2,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
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Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([],1)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> [2,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
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Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([],1)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> [2,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001117
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Values
([(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)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001318
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(load all 3 compositions to match this statistic)
Values
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Matching statistic: St001321
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(load all 3 compositions to match this statistic)
Values
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
Description
The number of vertices of the largest induced subforest of a graph.
The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St000053The number of valleys of the Dyck path. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000741The Colin de Verdière graph invariant. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000778The metric dimension of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for 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 $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001270The bandwidth of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001734The lettericity of a graph. St001746The coalition number of a graph. St001962The proper pathwidth of a graph. St000015The number of peaks of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000784The maximum of the length and the largest part of the integer partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001203We associate to 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 Dyck path as follows:
St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001345The Hamming dimension of a graph. St001638The book thickness of a graph. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001883The mutual visibility number of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000946The sum of the skew hook positions in a Dyck path. 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 $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001875The number of simple modules with projective dimension at most 1. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001644The dimension of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000907The number of maximal antichains of minimal length in a poset. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000327The number of cover relations in a poset. St000422The energy of a graph, if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset.
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