Your data matches 111 different statistics following compositions of up to 3 maps.
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Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000507: Standard tableaux ⟶ 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]]
=> 2 = 1 + 1
[2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [[1,3],[2]]
=> 2 = 1 + 1
[3,1,2] => [[1,2],[3]]
=> 2 = 1 + 1
[3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,3,4,2] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,4,2,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[2,3,1,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,3,4,1] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,4,1,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[2,4,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[3,1,4,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[3,2,4,1] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[3,4,1,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,1,2,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3 = 2 + 1
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> 3 = 2 + 1
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 3 = 2 + 1
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3 = 2 + 1
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 3 = 2 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
Description
The number of valleys of the Dyck path.
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [[1]]
=> 0
[1,2] => [[1,2]]
=> [[1],[2]]
=> 1
[2,1] => [[1],[2]]
=> [[1,2]]
=> 0
[1,2,3] => [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,1,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[3,2,1] => [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[1,2,3,4] => [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[2,3,1,4] => [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,3,4,1] => [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,4,1,3] => [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[2,4,3,1] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[3,1,2,4] => [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[4,2,1,3] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[4,3,1,2] => [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00066: Permutations inversePermutations
Mp00204: Permutations LLPSInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1 = 0 + 1
[1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[2,1] => [2,1] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [3,1,2] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,4,1,3] => [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[2,4,3,1] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,3,1,4] => [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [4,3,1,2] => [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [2,3,4,1] => [2,1,1]
=> 3 = 2 + 1
[4,1,3,2] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[4,2,1,3] => [3,2,4,1] => [3,1]
=> 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [3,1]
=> 2 = 1 + 1
[4,3,1,2] => [3,4,2,1] => [3,1]
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,5,2,4,3] => [3,1,1]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,2,4] => [3,1,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
Description
The length of the partition.
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00141: Binary trees pruning number to logarithmic heightDyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000211
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000211: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> {{1}}
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000319
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00204: Permutations LLPSInteger partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1]
=> 0
[1,2] => [.,[.,.]]
=> [2,1] => [2]
=> 1
[2,1] => [[.,.],.]
=> [1,2] => [1,1]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 3
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001581The achromatic number of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000362The size of a minimal vertex cover of a graph. St001304The number of maximally independent sets of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000527The width of the poset. St000167The number of leaves of an ordered tree. St000632The jump number of the poset. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000809The reduced reflection length of the permutation. St000702The number of weak deficiencies of a permutation. St000662The staircase size of the code of a permutation. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000308The height of the tree associated to a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St000141The maximum drop size of a permutation. St000292The number of ascents of a binary word. St000332The positive inversions of an alternating sign matrix. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000542The number of left-to-right-minima of a permutation. St000021The number of descents of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St000746The number of pairs with odd minimum in a perfect matching. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001935The number of ascents in a parking function. St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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 $eA$.