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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001960
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [3,2,1] => 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [1,3,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,1,2] => [2,1,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [4,2,3,1] => 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [4,1,2,3] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [1,4,2,3] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [4,2,1,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,2,4,1] => [2,1,4,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => [1,4,3,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => [2,4,3,1] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => [1,2,3,4] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => [4,3,2,1] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,1,2,3] => [3,4,2,1] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [5,1,2,3,4] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,5,2,3,4] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [5,2,1,3,4] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [5,2,3,1,4] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,2,4,1,5] => [2,1,5,3,4] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [1,5,3,2,4] => 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,2,4,5,1] => [2,5,3,1,4] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [5,3,2,1,4] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [5,2,1,4,3] => 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => [2,5,3,4,1] => 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => [1,2,3,5,4] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [2,4,1,3,5] => [5,3,2,4,1] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,1,4,5,2] => [1,2,3,4,5] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [2,4,1,5,3] => [1,5,3,4,2] => 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [2,3,5,1,4] => [5,2,4,3,1] => 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,3,2,5,1] => [3,2,1,5,4] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,4,5,2,1] => [2,3,1,4,5] => 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,5,4,3,1] => [1,5,4,3,2] => 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,4,5,1,2] => [2,3,4,1,5] => 0
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,5,4,1,3] => [5,4,3,2,1] => 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,1,2,3,5] => [3,5,2,4,1] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,5,3] => [3,5,4,1,2] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,5,2,4] => [1,2,4,5,3] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,5,1,3,4] => [5,4,2,3,1] => 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [4,5,2,3,1] => 1
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: St001713
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 0 + 2
[[1],[2]]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 0 + 2
[[1,2,3]]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3 = 1 + 2
[[1,3],[2]]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 0 + 2
[[1,2],[3]]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3 = 1 + 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 0 + 2
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 2
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 2
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 2
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 2
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 3 + 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 0 + 2
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 3 + 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 2
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 2
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 0 + 4
[[1],[2]]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 0 + 4
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 1 + 4
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 4 = 0 + 4
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 1 + 4
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 4 = 0 + 4
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 1 + 4
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 4
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 1 + 4
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 4
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 2 + 4
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 1 + 4
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 1 + 4
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 2 + 4
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 4
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 4
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 1 + 4
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 4
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 1 + 4
Description
The pebbling number of a connected graph.
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