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Your data matches 67 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000389
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000566
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [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]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 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] => ?
=> ? = 3
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St001251
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [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]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 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] => ?
=> ? = 3
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [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]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 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] => ?
=> ? = 3
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [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]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 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] => ?
=> ? = 3
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001657
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [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]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 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] => ?
=> ? = 3
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000345
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [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]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [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]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [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]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [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]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [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]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3 + 1
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2 + 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1 + 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2 + 1
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2 + 1
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2 + 1
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2 + 1
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2 + 1
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 2 + 1
[[[[[.,.],.],[.,.]],.],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3 + 1
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2 + 1
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2 + 1
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3 + 1
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2 + 1
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1 + 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2 + 1
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3 + 1
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2 + 1
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3 + 1
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2 + 1
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5 + 1
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5 + 1
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5 + 1
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5 + 1
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6 + 1
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6 + 1
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6 + 1
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6 + 1
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6 + 1
Description
The number of refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [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]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [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]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [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]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [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]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [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]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3 + 1
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2 + 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 1 + 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2 + 1
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2 + 1
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2 + 1
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2 + 1
[[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2 + 1
[[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 2 + 1
[[[[[.,.],.],[.,.]],.],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3 + 1
[[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2 + 1
[[.,[[[.,.],.],[[.,.],.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2 + 1
[[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3 + 1
[[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2 + 1
[[.,[[.,[.,[[[.,.],.],.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 1 + 1
[[.,[[[[[.,.],.],.],[.,.]],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2 + 1
[[[.,[[.,.],[[.,.],.]]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3 + 1
[[[[.,.],[.,[[[.,.],.],.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 2 + 1
[[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3 + 1
[[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2 + 1
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8] => ?
=> ? = 5 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8] => ?
=> ? = 5 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8] => ?
=> ? = 5 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],.]]
=> [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] => ?
=> ? = 5 + 1
[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [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] => ?
=> ? = 5 + 1
[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]
=> [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] => ?
=> ? = 5 + 1
[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [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] => ?
=> ? = 5 + 1
[[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,8,3,9,5,11,7,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,6,2,7,3,9,5,11,8,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,8,3,9,6,11,7,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,9,6,11,8,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,8,4,9,5,11,7,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,6,2,7,4,9,5,11,8,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,8,4,9,6,11,7,12,10] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,9,6,11,8,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,6,2,7,3,9,5,10,8,12,11] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,6,2,7,4,9,5,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,8,6,11,9,12,10] => ?
=> ? = 6 + 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [4,1,5,2,7,3,8,6,10,9,12,11] => ?
=> ? = 6 + 1
[[.,.],[[[.,.],[[.,.],[[.,.],[.,.]]]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,7,4,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,9,7,10,8,12,11] => ?
=> ? = 6 + 1
[[.,.],[[.,.],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,7,5,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,7,5,9,6,10,8,12,11] => ?
=> ? = 6 + 1
[[[.,.],[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,9,7,10,8,12,11] => ?
=> ? = 6 + 1
[[.,.],[[[.,.],[.,.]],[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,9,7,10,8,12,11] => ?
=> ? = 6 + 1
[[.,.],[[.,.],[[.,.],[[[.,.],[.,.]],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,9,7,10,8,12,11] => ?
=> ? = 6 + 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001389The number of partitions of the same length below the given integer partition. St000142The number of even parts of a partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000157The number of descents of a standard tableau. St000919The number of maximal left branches of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000507The number of ascents of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000251The number of nonsingleton blocks of a set partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000742The number of big ascents of a permutation after prepending zero. St001354The number of series nodes in the modular decomposition of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000834The number of right outer peaks of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000703The number of deficiencies of a permutation. St000386The number of factors DDU in a Dyck path. St000662The staircase size of the code of a permutation. St000245The number of ascents of a permutation. St001427The number of descents of a signed permutation. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001469The holeyness of a permutation. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St001896The number of right descents of a signed permutations. St001597The Frobenius rank of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001946The number of descents in a parking function. St000920The logarithmic height of a Dyck path.
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