Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000008
Mp00305: Permutations parking functionParking functions
Mp00319: Parking functions to compositionInteger compositions
Mp00039: Integer compositions complementInteger compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,2,1,1] => 11
[1,3,2] => [1,3,2] => [1,3,2] => [2,1,2,1] => 10
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,1,1] => 10
[2,3,1] => [2,3,1] => [2,3,1] => [1,2,1,2] => 8
[3,1,2] => [3,1,2] => [3,1,2] => [1,1,3,1] => 8
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,2,2] => 7
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000304
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
Mp00001: Alternating sign matrices to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
St000304: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [[1]]
=> [1] => 0
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => 2
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => 11
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => 10
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => 10
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => 8
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => 8
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => 7
Description
The load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.
St001168: Permutations ⟶ ℤResult quality: 86% values known / values provided: 89%distinct values known / distinct values provided: 86%
Values
[1] => ? = 0 + 3
[1,2] => 5 = 2 + 3
[2,1] => 4 = 1 + 3
[1,2,3] => 14 = 11 + 3
[1,3,2] => 13 = 10 + 3
[2,1,3] => 13 = 10 + 3
[2,3,1] => 11 = 8 + 3
[3,1,2] => 11 = 8 + 3
[3,2,1] => 10 = 7 + 3
Description
The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00160: Permutations graph of inversionsGraphs
St000467: Graphs ⟶ ℤResult quality: 43% values known / values provided: 44%distinct values known / distinct values provided: 43%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2
[2,1] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 10
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {7,8,8,11}
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {7,8,8,11}
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 10
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {7,8,8,11}
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {7,8,8,11}
Description
The hyper-Wiener index of a connected graph. This is $$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$
Matching statistic: St000112
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
Mp00001: Alternating sign matrices to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00107: Semistandard tableaux catabolismSemistandard tableaux
St000112: Semistandard tableaux ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 43%
Values
[1] => [[1]]
=> [[1]]
=> [[1]]
=> 0
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> [[1,1,2]]
=> 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> [[1,2,2]]
=> 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [[1,1,1,2,2],[3]]
=> ? ∊ {7,8,8,10,10,11}
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [[1,1,1,2,3],[3]]
=> ? ∊ {7,8,8,10,10,11}
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [[1,1,2,2,2],[3]]
=> ? ∊ {7,8,8,10,10,11}
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [[1,1,2,3,3],[3]]
=> ? ∊ {7,8,8,10,10,11}
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [[1,2,2,2,3],[3]]
=> ? ∊ {7,8,8,10,10,11}
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [[1,2,2,3,3],[3]]
=> ? ∊ {7,8,8,10,10,11}
Description
The sum of the entries reduced by the index of their row in a semistandard tableau. This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00160: Permutations graph of inversionsGraphs
St000464: Graphs ⟶ ℤResult quality: 29% values known / values provided: 33%distinct values known / distinct values provided: 29%
Values
[1] => [1,0]
=> [1] => ([],1)
=> ? = 0
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2
[2,1] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 10
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {7,8,8,11}
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {7,8,8,11}
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 10
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {7,8,8,11}
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {7,8,8,11}
Description
The Schultz index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} (d(u)+d(v))d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the Schultz index is related to the Wiener index via $S(T)=4W(T)-n(n-1)$ [2].
Matching statistic: St001684
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
Mp00001: Alternating sign matrices to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
St001684: Permutations ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 43%
Values
[1] => [[1]]
=> [[1]]
=> [1] => 0
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => 2
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => ? ∊ {7,8,8,10,10,11}
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => ? ∊ {7,8,8,10,10,11}
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => ? ∊ {7,8,8,10,10,11}
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => ? ∊ {7,8,8,10,10,11}
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => ? ∊ {7,8,8,10,10,11}
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => ? ∊ {7,8,8,10,10,11}
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].
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St001960: Permutations ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 43%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 0
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => ? ∊ {7,8,8,10,10,11}
[1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => ? ∊ {7,8,8,10,10,11}
[2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => ? ∊ {7,8,8,10,10,11}
[2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => ? ∊ {7,8,8,10,10,11}
[3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => ? ∊ {7,8,8,10,10,11}
[3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => ? ∊ {7,8,8,10,10,11}
Description
The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000077
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
Mp00082: Standard tableaux to Gelfand-Tsetlin patternGelfand-Tsetlin patterns
St000077: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 43%
Values
[1] => [1,0]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> 3 = 2 + 1
[2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {7,8,8,10,10,11} + 1
[1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {7,8,8,10,10,11} + 1
[2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {7,8,8,10,10,11} + 1
[2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {7,8,8,10,10,11} + 1
[3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {7,8,8,10,10,11} + 1
[3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {7,8,8,10,10,11} + 1
Description
The number of boxed and circled entries. An entry $a_{i,j}$ is boxed and circled if $a_{i-1,j-i} = a_{i,j} = a_{i-1,j}$.
Matching statistic: St001207
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St001207: Permutations ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 43%
Values
[1] => [1,0]
=> [[1],[2]]
=> [2,1] => 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 2 = 1 + 1
[2,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ? ∊ {7,8,8,10,10,11} + 1
[1,3,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => ? ∊ {7,8,8,10,10,11} + 1
[2,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => ? ∊ {7,8,8,10,10,11} + 1
[2,3,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => ? ∊ {7,8,8,10,10,11} + 1
[3,1,2] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? ∊ {7,8,8,10,10,11} + 1
[3,2,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? ∊ {7,8,8,10,10,11} + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
The following 1 statistic also match your data. Click on any of them to see the details.
St001645The pebbling number of a connected graph.