Loading [MathJax]/jax/output/HTML-CSS/jax.js

Your data matches 322 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000288
(load all 22 compositions to match this statistic)
Mapping
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 1 = 2 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 2 = 3 - 1
100 => 1 = 2 - 1
101 => 2 = 3 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St001372
(load all 15 compositions to match this statistic)
Mapping
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 1 = 2 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 2 = 3 - 1
100 => 1 = 2 - 1
101 => 2 = 3 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 1
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 2
1 => 1 => 1
00 => 00 => 3
01 => 01 => 2
10 => 01 => 2
11 => 11 => 1
000 => 000 => 4
001 => 001 => 3
010 => 001 => 3
011 => 011 => 2
100 => 001 => 3
101 => 011 => 2
110 => 011 => 2
111 => 111 => 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 0 = 1 - 1
01 => 10 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 0 = 1 - 1
001 => 100 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 110 => 2 = 3 - 1
100 => 100 => 1 = 2 - 1
101 => 110 => 2 = 3 - 1
110 => 110 => 2 = 3 - 1
111 => 111 => 3 = 4 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000392
(load all 7 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 0 = 1 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 0 = 1 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 2 = 3 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 2 = 3 - 1
110 => 011 => 2 = 3 - 1
111 => 111 => 3 = 4 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000753
(load all 6 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000753: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 0 = 1 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 0 = 1 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 2 = 3 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 2 = 3 - 1
110 => 011 => 2 = 3 - 1
111 => 111 => 3 = 4 - 1
Description
The Grundy value for the game of Kayles on a binary word. Two players alternately may remove either a single 1 or two adjacent 1's. The player facing the word which has only 0's looses.
Matching statistic: St000877
(load all 2 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 0 = 1 - 1
00 => 00 => 2 = 3 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 0 = 1 - 1
000 => 000 => 3 = 4 - 1
001 => 001 => 2 = 3 - 1
010 => 001 => 2 = 3 - 1
011 => 011 => 1 = 2 - 1
100 => 001 => 2 = 3 - 1
101 => 011 => 1 = 2 - 1
110 => 011 => 1 = 2 - 1
111 => 111 => 0 = 1 - 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St001419
(load all 8 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 0 = 1 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 0 = 1 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 2 = 3 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 2 = 3 - 1
110 => 011 => 2 = 3 - 1
111 => 111 => 3 = 4 - 1
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1
1 => [1,1] => [1,1]
 => 2
00 => [3] => [3]
 => 1
01 => [2,1] => [2,1]
 => 2
10 => [1,2] => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => 3
000 => [4] => [4]
 => 1
001 => [3,1] => [3,1]
 => 2
010 => [2,2] => [2,2]
 => 2
011 => [2,1,1] => [2,1,1]
 => 3
100 => [1,3] => [3,1]
 => 2
101 => [1,2,1] => [2,1,1]
 => 3
110 => [1,1,2] => [2,1,1]
 => 3
111 => [1,1,1,1] => [1,1,1,1]
 => 4
Description
The length of the partition.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 312 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000015The number of peaks of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of 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. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call 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. 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: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000391The sum of the positions of the ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000536The pathwidth of a graph. St000691The number of changes of a binary word. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001480The number of simple summands of the module J^2/J^3. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001721The degree of a binary word. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000087The number of induced subgraphs. St000105The number of blocks in the set partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000228The size of a partition. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000390The number of runs of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000668The least common multiple of the parts of the partition. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000708The product of the parts of an integer partition. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000843The decomposition number of a perfect matching. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001726The number of visible inversions of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001949The rigidity index of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000067The inversion number of the alternating sign matrix. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils of a permutation. St000358The number of occurrences of the pattern 31-2. St000377The dinv defect of an integer partition. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000502The number of successions of a set partitions. St000519The largest length of a factor maximising the subword complexity. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000728The dimension of a set partition. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000778The metric dimension of a graph. St000795The mad of a permutation. St000796The stat' of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000864The number of circled entries of the shifted recording tableau of a permutation. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000957The number of Bruhat lower covers of a permutation. 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 $c_{n-1}$. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001270The bandwidth of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001280The number of 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001486The number of corners of the ribbon associated with an integer composition. St001489The maximum of the number of descents and the number of inverse descents. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. 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. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001948The number of augmented double ascents of a permutation. St001962The proper pathwidth of a graph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. 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 $eA$. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001545The second Elser number of a connected graph. 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.