searching the database
Your data matches 100 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001115
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 0
[1,3,4,5,2] => 1
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 0
Description
The number of even descents of a permutation.
Matching statistic: St001092
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001092: Integer partitions ⟶ 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]
=> 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> []
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[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]
=> [2,2]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[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]
=> [2,2]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[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]
=> [4,3,2,1]
=> 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 0
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St001151
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> {{1,2}}
=> 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 0 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 0 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
Description
The number of blocks with odd minimum.
See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St001524
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? => ? = 0
[1,2] => 0 => 0 => 0 => 0
[2,1] => 1 => 1 => 1 => 0
[1,2,3] => 00 => 01 => 10 => 0
[1,3,2] => 01 => 10 => 01 => 0
[2,1,3] => 10 => 00 => 00 => 1
[2,3,1] => 01 => 10 => 01 => 0
[3,1,2] => 10 => 00 => 00 => 1
[3,2,1] => 11 => 11 => 11 => 1
[1,2,3,4] => 000 => 011 => 101 => 1
[1,2,4,3] => 001 => 101 => 110 => 0
[1,3,2,4] => 010 => 000 => 000 => 1
[1,3,4,2] => 001 => 101 => 110 => 0
[1,4,2,3] => 010 => 000 => 000 => 1
[1,4,3,2] => 011 => 110 => 011 => 0
[2,1,3,4] => 100 => 010 => 001 => 0
[2,1,4,3] => 101 => 100 => 010 => 1
[2,3,1,4] => 010 => 000 => 000 => 1
[2,3,4,1] => 001 => 101 => 110 => 0
[2,4,1,3] => 010 => 000 => 000 => 1
[2,4,3,1] => 011 => 110 => 011 => 0
[3,1,2,4] => 100 => 010 => 001 => 0
[3,1,4,2] => 101 => 100 => 010 => 1
[3,2,1,4] => 110 => 001 => 100 => 0
[3,2,4,1] => 101 => 100 => 010 => 1
[3,4,1,2] => 010 => 000 => 000 => 1
[3,4,2,1] => 011 => 110 => 011 => 0
[4,1,2,3] => 100 => 010 => 001 => 0
[4,1,3,2] => 101 => 100 => 010 => 1
[4,2,1,3] => 110 => 001 => 100 => 0
[4,2,3,1] => 101 => 100 => 010 => 1
[4,3,1,2] => 110 => 001 => 100 => 0
[4,3,2,1] => 111 => 111 => 111 => 1
[1,2,3,4,5] => 0000 => 0111 => 1011 => 1
[1,2,3,5,4] => 0001 => 1011 => 1101 => 1
[1,2,4,3,5] => 0010 => 0001 => 1000 => 1
[1,2,4,5,3] => 0001 => 1011 => 1101 => 1
[1,2,5,3,4] => 0010 => 0001 => 1000 => 1
[1,2,5,4,3] => 0011 => 1101 => 1110 => 1
[1,3,2,4,5] => 0100 => 0100 => 0010 => 1
[1,3,2,5,4] => 0101 => 1000 => 0100 => 1
[1,3,4,2,5] => 0010 => 0001 => 1000 => 1
[1,3,4,5,2] => 0001 => 1011 => 1101 => 1
[1,3,5,2,4] => 0010 => 0001 => 1000 => 1
[1,3,5,4,2] => 0011 => 1101 => 1110 => 1
[1,4,2,3,5] => 0100 => 0100 => 0010 => 1
[1,4,2,5,3] => 0101 => 1000 => 0100 => 1
[1,4,3,2,5] => 0110 => 0010 => 0001 => 1
[1,4,3,5,2] => 0101 => 1000 => 0100 => 1
[1,4,5,2,3] => 0010 => 0001 => 1000 => 1
[1,4,5,3,2] => 0011 => 1101 => 1110 => 1
Description
The degree of symmetry of a binary word.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Matching statistic: St001114
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001114: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => ? = 0
[1,2] => [1] => ? ∊ {0,0}
[2,1] => [1] => ? ∊ {0,0}
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 0
[1,3,4,2] => [1,3,2] => 0
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 0
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 0
[2,3,4,1] => [2,3,1] => 0
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 0
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 0
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 0
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 0
[1,3,2,5,4] => [1,3,2,4] => 0
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 0
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 0
[1,4,2,5,3] => [1,4,2,3] => 0
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 0
[1,4,5,3,2] => [1,4,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => 0
[1,5,2,4,3] => [1,2,4,3] => 1
Description
The number of odd descents of a permutation.
Matching statistic: St000183
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> []
=> 0
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000621
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,3,2,4] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,4,2,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,4,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,1,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,4,5,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,5,2,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,5,3,2,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,5,3,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,5,4,2,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,5,4,3,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,4,3,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,5,3,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,3,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,4,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,4,5,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,3,5,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,5,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[2,4,1,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,4,3,1,5] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,4,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,4,5,1,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,5,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[2,5,1,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,5,3,1,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,5,3,4,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,5,4,1,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,5,4,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[3,1,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[3,1,4,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[3,1,5,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[3,2,1,4,5] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,5,4] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,4,1,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,4,5,1] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[3,2,5,1,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,4,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[3,4,1,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,1,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,5,1,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,2,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,2,1,3,5] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St001031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000620
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000938
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
The following 90 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000941The number of characters of the symmetric group whose value on the partition is even. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001498The normalised height of a Nakayama algebra with magnitude 1. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000946The sum of the skew hook positions in a Dyck path. St000260The radius of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000137The Grundy value of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000706The product of the factorials of the multiplicities of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St000456The monochromatic index of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St001820The size of the image of the pop stack sorting operator. St001570The minimal number of edges to add to make a graph Hamiltonian. St001621The number of atoms of a lattice. St000944The 3-degree of an integer partition. St001624The breadth of a lattice. St001875The number of simple modules with projective dimension at most 1. St001895The oddness of a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!