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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St000445
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000878
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => => => ? = 1 - 1
[1,0,1,0]
=> [1,2] => 0 => 1 => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => 1 => 0 => -1 = 0 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 11 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 10 => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 01 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 10 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => 01 => 10 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 110 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 110 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 001 => 110 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 011 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 010 => -1 = 0 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 110 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 001 => 110 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 010 => 101 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 011 => 100 => -1 = 0 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 001 => 110 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 001 => 110 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 1111 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1110 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1101 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1110 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0001 => 1110 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1011 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1101 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1110 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0001 => 1110 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0010 => 1101 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0011 => 1100 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0001 => 1110 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0001 => 1110 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 0111 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 0110 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 0101 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 0110 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1001 => 0110 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1011 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1101 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1110 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0001 => 1110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0010 => 1101 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0011 => 1100 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0001 => 1110 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0001 => 1110 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0100 => 1011 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ? => ? => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ? => ? => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ? => ? => ? = 4 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ? => ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ? => ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ? => ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ? => ? => ? = 2 - 1
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000148
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ? = 0
Description
The number of odd parts of a partition.
Matching statistic: St000475
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ? = 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001247
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ? = 0
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ? = 0
Description
Sum of the odd parts of a partition.
Matching statistic: St000835
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000835: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000835: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ?
=> ? = 0
Description
The minimal difference in size when partitioning the integer partition into two subpartitions.
This is the optimal value of the optimisation version of the partition problem [1].
Matching statistic: St000992
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ?
=> ? = 0
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St001055
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001055: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001055: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2,1] => [2]
=> [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [3]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [2,2]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ?
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ?
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ?
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ?
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ?
=> ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ?
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ?
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ?
=> ? = 0
[1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [4,1,7,2,8,3,9,5,10,6] => ?
=> ?
=> ? = 0
Description
The Grundy value for the game of removing cells of a row in an integer partition.
Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.
Matching statistic: St000288
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1]
=> 1 => 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 11 => 2
[1,1,0,0]
=> [2,1] => [2]
=> 0 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 111 => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 01 => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 01 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 01 => 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 011 => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 011 => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 011 => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 011 => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 011 => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 00 => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 011 => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 011 => 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 011 => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 011 => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 00 => 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 011 => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 011 => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 0111 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 0111 => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 0111 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 0111 => 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? => ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? => ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? => ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? => ? = 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? => ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? => ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ?
=> ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? => ? = 4
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? => ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? => ? = 4
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ? => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? => ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? => ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? => ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? => ? = 4
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? => ? = 4
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,7,8,1,9,3] => ?
=> ? => ? = 5
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [6,9,1,2,3,4,5,7,8] => ?
=> ? => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? = 10
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,7,6,10,8,9] => ?
=> ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8] => ?
=> ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8] => ?
=> ? => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8] => ?
=> ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,3,2,6,4,8,5,10,7,9] => ?
=> ? => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2,5,4,8,6,10,7,9] => ?
=> ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,8,7,10,9] => ?
=> ? => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [2,1,6,3,7,4,8,5,10,9] => ?
=> ? => ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,8,5,10,9] => ?
=> ? => ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,8,5,10,9] => ?
=> ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,8,6,10,9] => ?
=> ? => ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,8,6,10,9] => ?
=> ? => ? = 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9] => ?
=> ? => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,8,6,10,9] => ?
=> ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,4,3,8,5,9,6,10,7] => ?
=> ? => ? = 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,4,2,8,5,9,6,10,7] => ?
=> ? => ? = 0
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,10,7] => ?
=> ? => ? = 0
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,5,3,8,4,9,6,10,7] => ?
=> ? => ? = 0
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,10,7] => ?
=> ? => ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,10,7] => ?
=> ? => ? = 0
[1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,8,3,9,4,10,7] => ?
=> ? => ? = 0
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000248The number of anti-singletons of a set partition. St000696The number of cycles in the breakpoint graph of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000247The number of singleton blocks of a set partition. St000215The number of adjacencies of a permutation, zero appended. St000385The number of vertices with out-degree 1 in a binary tree. St000022The number of fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000389The number of runs of ones of odd length in a binary word. St001524The degree of symmetry of a binary word. St001948The number of augmented double ascents of a permutation.
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