Your data matches 494 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001692
(load all 3 compositions to match this statistic)
Mapping
St001692: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 0
([],3)
 => 0
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 0
Description
The number of vertices with higher degree than the average degree in a graph.
Matching statistic: St001111
(load all 5 compositions to match this statistic)
Mapping
St001111: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1 = 0 + 1
([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
Description
The weak 2-dynamic chromatic number of a graph. A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-weak-dynamic number of a graph.
Matching statistic: St001112
(load all 4 compositions to match this statistic)
Mapping
St001112: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1 = 0 + 1
([],2)
 => 1 = 0 + 1
([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
Description
The 3-weak dynamic number of a graph. A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-weak-dynamic number of a graph.
Matching statistic: St000749
(load all 8 compositions to match this statistic)
Mapping
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 0
([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => [2]
 => 0
([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => [2,1]
 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields $$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3. This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001214
(load all 6 compositions to match this statistic)
Mapping
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 0
([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => [2]
 => 0
([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => [2,1]
 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
Description
The aft of an integer partition. The aft is the size of the partition minus the length of the first row or column, whichever is larger. See also [[St000784]].
Matching statistic: St001382
(load all 13 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 0
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 0
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 2
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St001644
(load all 3 compositions to match this statistic)
Mapping
Mp00156: Graphs —line graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],0)
 => 0
([],2)
 => ([],0)
 => 0
([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],0)
 => 0
([(1,2)],3)
 => ([],1)
 => 0
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St001646
(load all 2 compositions to match this statistic)
Mapping
Mp00259: Graphs —vertex addition⟶ Graphs
St001646: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],2)
 => 0
([],2)
 => ([],3)
 => 0
([(0,1)],2)
 => ([(1,2)],3)
 => 0
([],3)
 => ([],4)
 => 0
([(1,2)],3)
 => ([(2,3)],4)
 => 1
([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
Description
The number of edges that can be added without increasing the maximal degree of a graph. This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on $n$ vertices, which has maximal degree $n-1$ and therefore has statistic $\binom{n-1}{2}$.
Matching statistic: St001795
(load all 3 compositions to match this statistic)
Mapping
Mp00203: Graphs —cone⟶ Graphs
St001795: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => 0
([],2)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
Description
The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).
Matching statistic: St001912
(load all 17 compositions to match this statistic)
Mapping
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 0
([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => [2]
 => 0
([],3)
 => [1,1,1]
 => 2
([(1,2)],3)
 => [2,1]
 => 0
([(0,2),(1,2)],3)
 => [2,2]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
Description
The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
The following 484 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000088The row sums of the character table of the symmetric group. St000184The size of the centralizer of any permutation of given cycle type. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000517The Kreweras number of an integer partition. St000531The leading coefficient of the rook polynomial of an 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. St000935The number of ordered refinements of an integer partition. St001349The number of different graphs obtained from the given graph by removing an edge. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001739The number of graphs with the same edge polytope as the given graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St000024The number of double up and double down steps of a Dyck path. St000377The dinv defect of an integer partition. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000741The Colin de Verdière graph invariant. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000778The metric dimension of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. 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. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001949The rigidity index of a graph. St000071The number of maximal chains in a poset. St000087The number of induced subgraphs. St000182The number of permutations whose cycle type is the given integer partition. St000309The number of vertices with even degree. St000346The number of coarsenings of a partition. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000527The width of the poset. St000537The cutwidth of a graph. St000617The number of global maxima of a Dyck path. St000636The hull number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000909The number of maximal chains of maximal size in a poset. St000926The clique-coclique number of a graph. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001268The size of the largest ordinal summand in the poset. St001270The bandwidth of a graph. St001281The normalized isoperimetric number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001399The distinguishing number of a poset. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001746The coalition number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001962The proper pathwidth of a graph. St001883The mutual visibility number of a graph. St000452The number of distinct eigenvalues of a graph. St000008The major index of the composition. St000052The number of valleys of a Dyck path not on the x-axis. St000157The number of descents of a standard tableau. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000445The number of rises of length 1 of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001402The number of separators in a permutation. St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001721The degree of a binary word. St001930The weak major index of a binary word. St001956The comajor index for set-valued two-row standard Young tableaux. St000048The multinomial of the parts of a partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000626The minimal period of a binary word. St000742The number of big ascents of a permutation after prepending zero. St000808The number of up steps of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000983The length of the longest alternating subword. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001220The width of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001463The number of distinct columns in the nullspace of a graph. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001874Lusztig's a-function for the symmetric group. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000824The sum of the number of descents and the number of recoils of a permutation. 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. St000938The number of zeros of the symmetric group character corresponding to the partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001645The pebbling number of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001626The number of maximal proper sublattices of a lattice. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000690The size of the conjugacy class of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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 $eA$. 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 $eA$. 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 $eA$. St000699The toughness times the least common multiple of 1,. St000095The number of triangles of a graph. St000145The Dyson rank of a partition. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000311The number of vertices of odd degree in a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000327The number of cover relations in a poset. St000362The size of a minimal vertex cover of a graph. St000379The number of Hamiltonian cycles in a graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000536The pathwidth of a graph. St000640The rank of the largest boolean interval in a poset. St000680The Grundy value for Hackendot on posets. St000944The 3-degree of an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. 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. St001479The number of bridges of a graph. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001668The number of points of the poset minus the width of the poset. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001797The number of overfull subgraphs of a graph. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000012The area of a Dyck path. St000053The number of valleys of the Dyck path. St000089The absolute variation of a composition. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000148The number of odd parts 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. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000228The 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. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000312The number of leaves in a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000351The determinant of the adjacency matrix of a graph. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000378The diagonal inversion number of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000422The energy of a graph, if it is integral. St000442The maximal area to the right of an up step of a Dyck path. St000459The hook length of the base cell of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000478Another weight of a partition according to Alladi. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000661The number of rises of length 3 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000766The number of inversions of an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000928The sum of the coefficients of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000936The number of even values 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. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St000995The largest even part of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001092The number of distinct even parts of a partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001271The competition number of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001480The number of simple summands of the module J^2/J^3. St001484The number of singletons of an integer partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001524The degree of symmetry of a binary word. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001673The degree of asymmetry of an integer composition. St001697The shifted natural comajor index of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001777The number of weak descents in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001827The number of two-component spanning forests of a graph. St001846The number of elements which do not have a complement in the lattice. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001955The number of natural descents for set-valued two row standard Young tableaux. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000090The variation of a composition. St000091The descent variation of a composition. St000117The number of centered tunnels of a Dyck path. St000292The number of ascents of a binary word. St000296The length of the symmetric border of a binary word. St000348The non-inversion sum of a binary word. St000389The number of runs of ones of odd length in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000674The number of hills 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000693The modular (standard) major index of a standard tableau. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000878The number of ones minus the number of zeros of a binary word. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. 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. 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. St001161The major index north count of a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001669The number of single rises in a Dyck path. St001910The height of the middle non-run of a Dyck path. St001931The weak major index of an integer composition regarded as a word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000455The second largest eigenvalue of a graph if it is integral. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001330The hat guessing number of a graph. St000017The number of inversions of a standard tableau. St000291The number of descents of a binary word. St000347The inversion sum of a binary word. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000628The balance of a binary word. St000629The defect of a binary word. St000681The Grundy value of Chomp on Ferrers diagrams. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001596The number of two-by-two squares inside a skew partition. St001712The number of natural descents of a standard Young tableau. St000782The indicator function of whether a given perfect matching is an L & P matching.