Your data matches 285 different statistics following compositions of up to 3 maps.
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Matching statistic: St000110
(load all 9 compositions to match this statistic)
Mapping
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[2,1,3,4] => 2
[2,1,4,3] => 4
[1,2,3,4,5] => 1
[1,2,4,3,5] => 2
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001464
(load all 26 compositions to match this statistic)
Mapping
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[2,1,3,4] => 2
[2,1,4,3] => 4
[1,2,3,4,5] => 1
[1,2,4,3,5] => 2
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St001684
(load all 9 compositions to match this statistic)
Mapping
St001684: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 2 = 3 - 1
[3,1,2] => 2 = 3 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 3 = 4 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => 1 = 2 - 1
Description
The reduced word complexity of a permutation. For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$. For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$. This statistic appears in [1, Question 6.1].
Matching statistic: St000468
(load all 3 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St000468: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => 2
Description
The Hosoya index of a graph. This is the total number of matchings in the graph.
Matching statistic: St000883
(load all 3 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => 3
[3,1,2] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 4
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 2
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001855
(load all 6 compositions to match this statistic)
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 3
[3,1,2] => [3,1,2] => 3
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 4
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 2
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St000004
(load all 9 compositions to match this statistic)
Mapping
Mp00067: Permutations —Foata bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [2,3,1] => 2 = 3 - 1
[3,1,2] => [1,3,2] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [4,2,1,3] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
Description
The major index of a permutation. This is the sum of the positions of its descents, $$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$ Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$. A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000305
(load all 10 compositions to match this statistic)
Mapping
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [2,3,1] => 1 = 2 - 1
[2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [3,1,2] => 2 = 3 - 1
[3,1,2] => [1,3,2] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[1,3,2,4] => [2,3,1,4] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [3,2,4,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [2,3,4,1,5] => 1 = 2 - 1
Description
The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000032
(load all 2 compositions to match this statistic)
Mapping
Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [1,0,1,0]
 => 1
[2,1] => [2,1] => [1,1,0,0]
 => 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 3
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000058
(load all 19 compositions to match this statistic)
Mapping
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [3,1,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => [2,3,1] => 3
[3,1,2] => [1,3,2] => [3,1,2] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 2
[1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [4,2,1,3] => [2,4,1,3] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 2
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
The following 275 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000071The number of maximal chains in a poset. St000228The size of a partition. St000300The number of independent sets of vertices of a graph. St000363The number of minimal vertex covers of a graph. St000909The number of maximal chains of maximal size in a poset. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001527The cyclic permutation representation number of an integer partition. St000008The major index of the composition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000209Maximum difference of elements in cycles. St000304The load of a permutation. St000330The (standard) major index of a standard tableau. St000670The reversal length of a permutation. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. 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. 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. St001671Haglund's hag of a permutation. St000054The first entry of the permutation. St000086The number of subgraphs. St000147The largest part of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000910The number of maximal chains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001093The detour number of a graph. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001110The 3-dynamic chromatic number of a graph. St001220The width of a permutation. St001268The size of the largest ordinal summand in the poset. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001497The position of the largest weak excedence of a permutation. St001555The order of a signed permutation. St001725The harmonious chromatic number of a graph. St001813The product of the sizes of the principal order filters in a poset. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000332The positive inversions of an alternating sign matrix. St000339The maf index of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000446The disorder of a permutation. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000867The sum of the hook lengths in the first row of an integer partition. St000868The aid statistic in the sense of Shareshian-Wachs. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001161The major index north count of a Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001375The pancake length of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001649The length of a longest trail in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001726The number of visible inversions of a permutation. St001759The Rajchgot index of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001869The maximum cut size of a graph. St001894The depth of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000100The number of linear extensions of a poset. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000794The mak of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001346The number of parking functions that give the same permutation. St001959The product of the heights of the peaks of a Dyck path. St000216The absolute length of a permutation. St000391The sum of the positions of the ones in a binary word. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000693The modular (standard) major index of a standard tableau. St000792The Grundy value for the game of ruler on a binary word. St000797The stat`` of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). 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. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000082The number of elements smaller than a binary tree in Tamari order. St000668The least common multiple of the parts of the partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001062The maximal size of a block of a set partition. St001330The hat guessing number of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St000289The decimal representation of a binary word. St000354The number of recoils of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000849The number of 1/3-balanced pairs in a poset. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001061The number of indices that are both descents and recoils of a permutation. St001480The number of simple summands of the module J^2/J^3. St001721The degree of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001209The pmaj statistic of a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001645The pebbling number of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000301The number of facets of the stable set polytope of a graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000939The number of characters of the symmetric group whose value on the partition is positive. St001391The disjunction number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001834The number of non-isomorphic minors of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000741The Colin de Verdière graph invariant. St001118The acyclic chromatic index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001441The number of non-empty connected induced subgraphs of a graph. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000766The number of inversions of an integer composition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000137The Grundy value of an integer partition. St000225Difference between largest and smallest parts in a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000707The product of the factorials of the parts. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001568The smallest positive integer that does not appear twice in the partition. St000072The number of circled entries. St000091The descent variation of a composition. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000929The constant term of the character polynomial of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St000467The hyper-Wiener index of a connected graph. St000749The 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. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001176The size of a partition minus its first part. St001961The sum of the greatest common divisors of all pairs of parts. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000646The number of big ascents of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000464The Schultz index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001545The second Elser number of a connected graph. St000264The girth of a graph, which is not a tree. St000302The determinant of the distance matrix of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000736The last entry in the first row of a semistandard tableau. St001060The distinguishing index of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. 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. 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. 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. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St000075The orbit size of a standard tableau under promotion. St000177The number of free tiles in the pattern. St000178Number of free entries. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001948The number of augmented double ascents of a permutation. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000928The sum of the coefficients of the character polynomial of an integer partition.