Your data matches 170 different statistics following compositions of up to 3 maps.
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Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000021: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000238: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The number of indices that are not small weak excedances. A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$.
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000240: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The number of indices that are not small excedances. A small excedance is an index $i$ for which $\pi_i = i+1$.
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000333: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000354: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000829: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The Ulam distance of a permutation to the identity permutation. This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$. In other words, this statistic plus [[St000062]] equals $n$.
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000831: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The number of indices that are either descents or recoils. This is, for a permutation $\pi$ of length $n$, this statistics counts the set $$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St001061: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The number of indices that are both descents and recoils of a permutation.
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St001489: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
Description
The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000325: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 2 = 1 + 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 4 = 3 + 1
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 3 = 2 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 6 = 5 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4 = 3 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 5 = 4 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 5 = 4 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 5 = 4 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 5 = 4 + 1
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
The following 160 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000470The number of runs in a permutation. St000670The reversal length of a permutation. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000245The number of ascents of a permutation. St000288The number of ones in a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000362The size of a minimal vertex cover of a graph. St000393The number of strictly increasing runs in a binary word. St000451The length of the longest pattern of the form k 1 2. St000482The (zero)-forcing number of a graph. St000495The number of inversions of distance at most 2 of a permutation. St000507The number of ascents of a standard tableau. St000519The largest length of a factor maximising the subword complexity. St000619The number of cyclic descents of a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000795The mad of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000922The minimal number such that all substrings of this length are unique. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001176The size of a partition minus its first part. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001427The number of descents of a signed permutation. St000004The major index of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000156The Denert index of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000301The number of facets of the stable set polytope of a graph. St000308The height of the tree associated to a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000742The number of big ascents of a permutation after prepending zero. St000794The mak of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000991The number of right-to-left minima of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001388The number of non-attacking neighbors of a permutation. St001554The number of distinct nonempty subtrees of a binary tree. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000438The position of the last up step in a Dyck path. St000133The "bounce" 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. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000792The Grundy value for the game of ruler on a binary word. St000833The comajor index of a permutation. St000837The number of ascents of distance 2 of a permutation. St001077The prefix exchange distance of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001960The number of descents of a permutation minus one if its first entry is not one. St001812The biclique partition number of a graph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000189The number of elements in the poset. St000528The height of a poset. St000907The number of maximal antichains 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. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000806The semiperimeter of the associated bargraph. St000893The number of distinct diagonal sums of an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000259The diameter of a connected graph. St000327The number of cover relations in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. 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. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000894The trace of an alternating sign matrix. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St001684The reduced word complexity of a permutation. St000077The number of boxed and circled entries. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001769The reflection length of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001875The number of simple modules with projective dimension at most 1. 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. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000422The energy of a graph, if it is integral. St000741The Colin de Verdière graph invariant. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001209The pmaj statistic of a parking function. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001894The depth of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.