Your data matches 298 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000088
(load all 4 compositions to match this statistic)
Mapping
St000088: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 2 = 3 - 1
[1,1]
 => 0 = 1 - 1
[3]
 => 3 = 4 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 1 = 2 - 1
Description
The row sums of the character table of the symmetric group. Equivalently, this is the multiplicity of the irreducible representation corresponding to the given partition in the adjoint representation of the symmetric group.
Matching statistic: St000148
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Mapping
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 2 = 3 - 1
[3]
 => 1 = 2 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 3 = 4 - 1
Description
The number of odd parts of a partition.
Matching statistic: St000835
(load all 5 compositions to match this statistic)
Mapping
St000835: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 2 = 3 - 1
[1,1]
 => 0 = 1 - 1
[3]
 => 3 = 4 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 1 = 2 - 1
Description
The minimal difference in size when partitioning the integer partition into two subpartitions. This is the optimal value of the optimisation version of the partition problem [1].
Matching statistic: St000992
(load all 4 compositions to match this statistic)
Mapping
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 2 = 3 - 1
[1,1]
 => 0 = 1 - 1
[3]
 => 3 = 4 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 1 = 2 - 1
Description
The alternating sum of the parts of an integer partition. For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St001055
(load all 5 compositions to match this statistic)
Mapping
St001055: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 2 = 3 - 1
[1,1]
 => 0 = 1 - 1
[3]
 => 3 = 4 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 1 = 2 - 1
Description
The Grundy value for the game of removing cells of a row in an integer partition. Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.
Matching statistic: St001247
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Mapping
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 2 = 3 - 1
[3]
 => 1 = 2 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 3 = 4 - 1
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St000288
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Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 1 = 2 - 1
[1,1,1]
 => 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: St000392
(load all 4 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 1 = 2 - 1
[1,1,1]
 => 111 => 3 = 4 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000445
(load all 9 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => 1 = 2 - 1
[2]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1]
 => [1,1,0,0]
 => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000753
(load all 4 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
St000753: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 1 = 2 - 1
[1,1,1]
 => 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.
The following 288 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001249Sum of the odd parts of a partition. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000145The Dyson rank of a partition. St000025The number of initial rises of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. 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. St000982The length of the longest constant subword. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. St001481The minimal height of a peak of a Dyck path. St000022The number of fixed points of a permutation. St000117The number of centered tunnels of a Dyck path. St000215The number of adjacencies of a permutation, zero appended. St000241The number of cyclical small excedances. St000297The number of leading ones in a binary word. St000338The number of pixed points of a permutation. St000389The number of runs of ones of odd length in a binary word. St000439The position of the first down step of a Dyck path. St000475The number of parts equal to 1 in a partition. St001008Number of indecomposable injective modules with projective dimension 1 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. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001777The number of weak descents in an integer composition. St001910The height of the middle non-run of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000054The first entry of the permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. 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. St000444The length of the maximal rise of a Dyck path. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000657The smallest part of an integer composition. 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. St000686The finitistic dominant dimension of a Dyck path. St000700The protection number of an ordered tree. St000702The number of weak deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000740The last entry of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000878The number of ones minus the number of zeros of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We 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 $c_{n−1}$. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001670The connected partition 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. St001733The number of weak left to right maxima of a Dyck path. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000089The absolute variation of a composition. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000153The number of adjacent cycles of a permutation. St000221The number of strong fixed points of a permutation. St000237The number of small exceedances. St000239The number of small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000385The number of vertices with out-degree 1 in a binary tree. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000618The number of self-evacuating tableaux of given shape. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000738The first entry in the last row of a standard tableau. St000792The Grundy value for the game of ruler on a binary word. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000819The propagating number of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. 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 $c_{n-1}$. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000989The number of final rises of a permutation. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. 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 $eA$. 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. 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. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001358The largest degree of a regular subgraph of a graph. St001480The number of simple summands of the module J^2/J^3. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001510The number of self-evacuating linear extensions of a finite poset. St001524The degree of symmetry of a binary word. St001586The number of odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001675The number of parts equal to the part in the reversed composition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001691The number of kings in a graph. St001812The biclique partition number of a graph. St001903The number of fixed points of a parking function. St001948The number of augmented double ascents of a permutation. St001962The proper pathwidth of a graph. St000090The variation of a composition. St000146The Andrews-Garvan crank of a partition. St000894The trace of an alternating sign matrix. St001377The major index minus the number of inversions of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000247The number of singleton blocks of a set partition. St000461The rix statistic of a permutation. St000674The number of hills of a Dyck path. St000248The number of anti-singletons of a set partition. St000454The largest eigenvalue of a graph if it is integral. St000680The Grundy value for Hackendot on posets. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000806The semiperimeter of the associated bargraph. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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 $eA$. St001378The product of the cohook lengths of the integer partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001383The BG-rank of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St000933The number of multipartitions of sizes given by 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. St001128The exponens consonantiae of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000762The sum of the positions of the weak records of an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000850The number of 1/2-balanced pairs in a poset. 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. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000302The determinant of the distance matrix of a connected graph. St000471The sum of the ascent tops of a permutation. St000673The number of non-fixed points of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement of a permutation. St000936The number of even values of the symmetric group character corresponding to the partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001964The interval resolution global dimension of a poset. St000652The maximal difference between successive positions of a permutation.