Your data matches 240 different statistics following compositions of up to 3 maps.
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Matching statistic: St001884
(load all 10 compositions to match this statistic)
Mapping
St001884: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 = 2 - 1
1 => 1 = 2 - 1
00 => 2 = 3 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 3 = 4 - 1
001 => 1 = 2 - 1
010 => 2 = 3 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 2 = 3 - 1
110 => 1 = 2 - 1
111 => 3 = 4 - 1
Description
The number of borders of a binary word. A border of a binary word $w$ is a word which is both a prefix and a suffix of $w$.
Matching statistic: St000295
(load all 10 compositions to match this statistic)
Mapping
St000295: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 2 - 2
1 => 0 = 2 - 2
00 => 1 = 3 - 2
01 => 0 = 2 - 2
10 => 0 = 2 - 2
11 => 1 = 3 - 2
000 => 2 = 4 - 2
001 => 0 = 2 - 2
010 => 1 = 3 - 2
011 => 0 = 2 - 2
100 => 0 = 2 - 2
101 => 1 = 3 - 2
110 => 0 = 2 - 2
111 => 2 = 4 - 2
Description
The length of the border of a binary word. The border of a word is the longest word which is both a proper prefix and a proper suffix, including a possible empty border.
Matching statistic: St000382
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 2 = 3 - 1
01 => [1,1] => 1 = 2 - 1
10 => [1,1] => 1 = 2 - 1
11 => [2] => 2 = 3 - 1
000 => [3] => 3 = 4 - 1
001 => [2,1] => 2 = 3 - 1
010 => [1,1,1] => 1 = 2 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 1 = 2 - 1
110 => [2,1] => 2 = 3 - 1
111 => [3] => 3 = 4 - 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 2 = 3 - 1
01 => [1,1] => 1 = 2 - 1
10 => [1,1] => 1 = 2 - 1
11 => [2] => 2 = 3 - 1
000 => [3] => 3 = 4 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 1 = 2 - 1
011 => [1,2] => 2 = 3 - 1
100 => [1,2] => 2 = 3 - 1
101 => [1,1,1] => 1 = 2 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 3 = 4 - 1
Description
The last part of an integer composition.
Matching statistic: St000757
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 2 = 3 - 1
100 => [1,2] => 2 = 3 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
Matching statistic: St000765
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 1 = 2 - 1
01 => [1,1] => 2 = 3 - 1
10 => [1,1] => 2 = 3 - 1
11 => [2] => 1 = 2 - 1
000 => [3] => 1 = 2 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 3 = 4 - 1
011 => [1,2] => 2 = 3 - 1
100 => [1,2] => 2 = 3 - 1
101 => [1,1,1] => 3 = 4 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 1 = 2 - 1
Description
The number of weak records in an integer composition. A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
Matching statistic: St000982
(load all 7 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 01 => 1 = 2 - 1
01 => 10 => 1 = 2 - 1
10 => 11 => 2 = 3 - 1
11 => 11 => 2 = 3 - 1
000 => 001 => 2 = 3 - 1
001 => 010 => 1 = 2 - 1
010 => 101 => 1 = 2 - 1
011 => 101 => 1 = 2 - 1
100 => 101 => 1 = 2 - 1
101 => 110 => 2 = 3 - 1
110 => 111 => 3 = 4 - 1
111 => 111 => 3 = 4 - 1
Description
The length of the longest constant subword.
Matching statistic: St001777
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 0 = 2 - 2
1 => [1] => 0 = 2 - 2
00 => [2] => 0 = 2 - 2
01 => [1,1] => 1 = 3 - 2
10 => [1,1] => 1 = 3 - 2
11 => [2] => 0 = 2 - 2
000 => [3] => 0 = 2 - 2
001 => [2,1] => 1 = 3 - 2
010 => [1,1,1] => 2 = 4 - 2
011 => [1,2] => 0 = 2 - 2
100 => [1,2] => 0 = 2 - 2
101 => [1,1,1] => 2 = 4 - 2
110 => [2,1] => 1 = 3 - 2
111 => [3] => 0 = 2 - 2
Description
The number of weak descents in an integer composition. A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 2
1 => [1] => [1,0]
 => 2
00 => [2] => [1,1,0,0]
 => 3
01 => [1,1] => [1,0,1,0]
 => 2
10 => [1,1] => [1,0,1,0]
 => 2
11 => [2] => [1,1,0,0]
 => 3
000 => [3] => [1,1,1,0,0,0]
 => 4
001 => [2,1] => [1,1,0,0,1,0]
 => 3
010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
011 => [1,2] => [1,0,1,1,0,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
110 => [2,1] => [1,1,0,0,1,0]
 => 3
111 => [3] => [1,1,1,0,0,0]
 => 4
Description
The position of the first down step of a Dyck path.
Matching statistic: St000469
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000469: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => 2
1 => [1,1] => ([(0,1)],2)
 => 2
00 => [3] => ([],3)
 => 3
01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
10 => [1,2] => ([(1,2)],3)
 => 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
000 => [4] => ([],4)
 => 4
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
100 => [1,3] => ([(2,3)],4)
 => 2
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
Description
The distinguishing number of a graph. This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring. For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
The following 230 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000918The 2-limited packing number of a graph. 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 $c_{n-1}$. St001366The maximal multiplicity of a degree of a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St000544The cop number of a graph. St000626The minimal period of a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000916The packing number of a graph. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001415The length of the longest palindromic prefix of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000377The dinv defect of an integer partition. St000691The number of changes of a binary word. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001479The number of bridges of a graph. St001826The maximal number of leaves on a vertex of a graph. St000054The first entry of the permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000738The first entry in the last row of a standard tableau. St000922The minimal number such that all substrings of this length are unique. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001486The number of corners of the ribbon associated with an integer composition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000160The multiplicity of the smallest part of a partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000654The first descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000740The last entry of a permutation. St000759The smallest missing part in an integer partition. St000767The number of runs in an integer composition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000877The depth of the binary word interpreted as a path. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of 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. St001050The number of terminal closers of a set partition. St001052The length of the exterior of a permutation. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001096The size of the overlap set 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. 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. St001286The annihilation number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001571The Cartan determinant of the integer partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination 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. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001933The largest multiplicity of a part in an integer partition. St001949The rigidity index of a graph. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000148The number of odd parts of a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000338The number of pixed points of a permutation. St000359The number of occurrences of the pattern 23-1. St000369The dinv deficit of a Dyck path. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000538The number of even inversions of a permutation. St000549The number of odd partial sums of an integer partition. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000735The last entry on the main diagonal of a standard tableau. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000836The number of descents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000989The number of final rises of a permutation. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001119The length of a shortest maximal path in a graph. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001252Half the sum of the even parts of a partition. St001274The number of indecomposable injective modules with projective dimension equal to two. St001357The maximal degree of a regular spanning subgraph of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001484The number of singletons of an integer partition. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001281The normalized isoperimetric number of a graph. St000061The number of nodes on the left branch of a binary tree. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000823The number of unsplittable factors of the set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001645The pebbling number of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000741The Colin de Verdière graph invariant. St000934The 2-degree of an integer partition. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001948The number of augmented double ascents of a permutation. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001644The dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001624The breadth of a lattice. St000259The diameter of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. 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 $eA$. 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 $eA$. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001545The second Elser number of a connected graph. St001060The distinguishing index of a graph. St000464The Schultz index of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. 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.