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Matching statistic: St000703
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 1
[3,1,2] => {{1,2,3}}
=> [2,3,1] => 1
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 2
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000994
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 1
[3,1,2] => {{1,2,3}}
=> [2,3,1] => 1
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 2
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000291
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 1 => 0
[1,2] => {{1},{2}}
=> [1,1] => 11 => 0
[2,1] => {{1,2}}
=> [2] => 10 => 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => 111 => 0
[1,3,2] => {{1},{2,3}}
=> [1,2] => 110 => 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => 101 => 1
[2,3,1] => {{1,2,3}}
=> [3] => 100 => 1
[3,1,2] => {{1,2,3}}
=> [3] => 100 => 1
[3,2,1] => {{1,3},{2}}
=> [2,1] => 101 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => 1110 => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => 1101 => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => 1100 => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => 1100 => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => 1101 => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => 1010 => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,1,3] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => 1000 => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => 1001 => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => 1010 => 2
[3,4,2,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,1,2,3] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => 1001 => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => 1001 => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
[4,3,1,2] => {{1,2,3,4}}
=> [4] => 1000 => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => 1010 => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 11110 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 11101 => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 11101 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 11011 => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => 11010 => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 11001 => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 11011 => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => 11001 => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => 11010 => 2
[2,1,4,3,6,5,8,7,10,9] => {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ? => ? => ? = 5
[2,3,4,5,6,7,8,9,1,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[2,1,3,4,5,6,7,8,9,10] => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,2,3,4,5,6,7,8,9,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 0
[9,8,7,6,5,4,3,2,1,10] => {{1,9},{2,8},{3,7},{4,6},{5},{10}}
=> [2,2,2,2,1,1] => 1010101011 => ? = 4
[1,10,9,8,7,6,5,4,3,2] => {{1},{2,10},{3,9},{4,8},{5,7},{6}}
=> [1,2,2,2,2,1] => 1101010101 => ? = 4
[10,2,1,3,4,5,6,7,8,9] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[3,2,4,5,6,7,8,9,10,1] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[9,1,2,3,4,5,6,7,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[8,9,1,2,3,4,5,6,7,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[2,4,1,6,3,8,5,9,7,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[3,2,1,4,5,6,7,8,9,10] => {{1,3},{2},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[3,4,5,6,7,8,9,1,2,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[3,1,5,2,7,4,9,6,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[6,7,8,9,1,2,3,4,5,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[5,6,7,8,9,1,2,3,4,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[9,7,5,3,1,8,6,4,2,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[3,5,7,9,1,2,4,6,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[8,6,4,2,9,1,3,5,7,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[9,6,4,2,7,1,3,5,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[9,7,5,3,1,2,4,6,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[9,8,7,6,4,3,2,1,5,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[5,6,4,7,3,8,2,9,1,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[3,1,9,6,2,8,4,5,7,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[7,4,1,6,2,3,9,5,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[9,4,1,8,7,2,3,5,6,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[9,6,5,1,2,8,3,4,7,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[8,2,1,10,4,5,6,7,3,9] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[4,2,6,8,10,1,3,5,7,9] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[8,2,9,3,10,4,5,6,7,1] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[4,2,5,6,7,8,9,10,1,3] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[7,2,8,9,10,1,3,4,5,6] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[3,2,8,9,10,1,4,5,6,7] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[3,2,6,1,4,7,8,9,10,5] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[7,6,4,1,8,5,2,9,3,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[7,2,1,8,3,9,4,10,5,6] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[4,3,7,2,1,5,8,9,6,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[4,6,8,3,2,1,9,7,5,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[8,2,6,10,9,5,4,3,7,1] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[7,2,8,3,9,4,10,5,1,6] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[7,2,8,9,3,4,10,1,5,6] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[7,2,8,9,10,3,4,1,5,6] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[5,9,8,7,6,4,3,2,1,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[4,3,6,2,7,8,1,9,5,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
[4,2,6,3,8,5,10,7,1,9] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[3,2,9,5,6,7,8,1,10,4] => {{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
[9,1,2,3,6,4,5,7,8,10] => {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
Description
The number of descents of a binary word.
Matching statistic: St001280
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Values
[1] => {{1}}
=> {{1}}
=> [1]
=> 0
[1,2] => {{1},{2}}
=> {{1},{2}}
=> [1,1]
=> 0
[2,1] => {{1,2}}
=> {{1,2}}
=> [2]
=> 1
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> 0
[1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> [2,1]
=> 1
[2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> 1
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> [3]
=> 1
[3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> [3]
=> 1
[3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> [2,1]
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [3,1]
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [3,1]
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,2]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[2,4,1,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [3,1]
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,1]
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [2,2]
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[4,1,3,2] => {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [3,1]
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,1]
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[4,3,1,2] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> [4,1]
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> [4,1]
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> [4,1]
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> 2
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ? = 1
[8,2,3,1,4,5,6,7] => {{1,4,5,6,7,8},{2},{3}}
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ? = 1
[7,6,4,3,5,2,1,8] => {{1,7},{2,6},{3,4},{5},{8}}
=> {{1,4},{2,6},{3,7},{5},{8}}
=> ?
=> ? = 3
[4,5,6,1,2,3,7,8] => {{1,4},{2,5},{3,6},{7},{8}}
=> {{1,6},{2,5},{3,4},{7},{8}}
=> ?
=> ? = 3
[3,5,8,7,6,4,2,1] => {{1,3,8},{2,4,5,6,7}}
=> {{1,5,7},{2,3,4,6,8}}
=> ?
=> ? = 2
[2,4,3,1,7,6,8,5] => {{1,2,4},{3},{5,7,8},{6}}
=> {{1,2,4},{3},{5,7,8},{6}}
=> ?
=> ? = 2
[2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6,7,8}}
=> ?
=> ? = 2
[1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> ?
=> ? = 2
[1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> ?
=> ? = 2
[1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ?
=> ? = 2
[1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ?
=> ? = 2
[1,8,7,4,6,5,3,2] => {{1},{2,8},{3,7},{4},{5,6}}
=> {{1},{2,6},{3,7},{4},{5,8}}
=> ?
=> ? = 3
[1,8,7,5,4,6,3,2] => {{1},{2,8},{3,7},{4,5},{6}}
=> {{1},{2,5},{3,7},{4,8},{6}}
=> ?
=> ? = 3
[7,6,3,5,4,2,1,8] => {{1,7},{2,6},{3},{4,5},{8}}
=> {{1,5},{2,6},{3},{4,7},{8}}
=> ?
=> ? = 3
[2,4,6,1,8,3,5,7] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ?
=> ? = 3
[2,4,6,1,7,3,8,5] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ?
=> ? = 3
[2,3,1,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6,7,8}}
=> ?
=> ? = 2
[3,1,2,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6,7,8}}
=> ?
=> ? = 2
[3,1,2,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6,7,8}}
=> ?
=> ? = 2
[1,4,5,6,7,2,3,8] => {{1},{2,4,6},{3,5,7},{8}}
=> {{1},{2,7},{3,4,5,6},{8}}
=> ?
=> ? = 2
[4,1,6,2,7,3,8,5] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ?
=> ? = 3
[4,1,6,2,8,3,5,7] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ?
=> ? = 3
[1,4,2,3,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> ?
=> ? = 2
[1,4,2,3,7,5,6,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> ?
=> ? = 2
[1,2,5,6,3,4,7,8] => {{1},{2},{3,5},{4,6},{7},{8}}
=> {{1},{2},{3,6},{4,5},{7},{8}}
=> ?
=> ? = 2
[1,6,7,2,3,4,5,8] => {{1},{2,4,6},{3,5,7},{8}}
=> {{1},{2,7},{3,4,5,6},{8}}
=> ?
=> ? = 2
[2,1,4,3,6,5,8,7,10,9] => {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ?
=> ? = 5
[6,7,8,9,10,1,2,3,4,5] => {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ?
=> ? = 5
[4,2,6,8,1,3,5,7] => {{1,4,5,7,8},{2},{3,6}}
=> {{1,5,6},{2},{3,4,7,8}}
=> ?
=> ? = 2
[5,3,2,6,1,4,7,8] => {{1,5},{2,3},{4,6},{7},{8}}
=> {{1,3},{2,6},{4,5},{7},{8}}
=> ?
=> ? = 3
[8,6,2,4,1,3,5,7] => {{1,5,7,8},{2,3,6},{4}}
=> {{1,3,5,6},{2,7,8},{4}}
=> ?
=> ? = 2
[6,4,3,2,7,1,5,8] => {{1,6},{2,4},{3},{5,7},{8}}
=> {{1,4},{2,7},{3},{5,6},{8}}
=> ?
=> ? = 3
[6,8,5,7,2,4,1,3] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[7,5,3,8,2,6,1,4] => {{1,7},{2,5},{3},{4,8},{6}}
=> {{1,8},{2,5},{3},{4,7},{6}}
=> ?
=> ? = 3
[8,4,1,5,6,7,2,3] => {{1,3,8},{2,4,5,6,7}}
=> {{1,5,7},{2,3,4,6,8}}
=> ?
=> ? = 2
[6,3,8,1,2,7,4,5] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[6,5,8,7,3,4,1,2] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[1,6,7,4,5,2,3,8] => {{1},{2,6},{3,7},{4},{5},{8}}
=> {{1},{2,7},{3,6},{4},{5},{8}}
=> ?
=> ? = 2
[4,3,5,6,8,7,1,2] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[6,2,8,4,5,1,7,3] => {{1,6},{2},{3,8},{4},{5},{7}}
=> {{1,8},{2},{3,6},{4},{5},{7}}
=> ?
=> ? = 2
[4,1,3,2,8,6,5,7] => {{1,2,4},{3},{5,7,8},{6}}
=> {{1,2,4},{3},{5,7,8},{6}}
=> ?
=> ? = 2
[4,5,2,7,8,1,6,3] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[1,5,3,7,2,6,4,8] => {{1},{2,5},{3},{4,7},{6},{8}}
=> {{1},{2,7},{3},{4,5},{6},{8}}
=> ?
=> ? = 2
[7,8,5,6,2,1,4,3] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[7,8,2,1,3,4,6,5] => {{1,4,6,7},{2,3,5,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> ?
=> ? = 2
[7,2,6,1,4,3,8,5] => {{1,4,5,7,8},{2},{3,6}}
=> {{1,5,6},{2},{3,4,7,8}}
=> ?
=> ? = 2
[4,2,6,7,8,3,5,1] => {{1,4,5,7,8},{2},{3,6}}
=> {{1,5,6},{2},{3,4,7,8}}
=> ?
=> ? = 2
[3,2,6,8,4,1,5,7] => {{1,3,6},{2},{4,5,7,8}}
=> {{1,3,5,6},{2},{4,7,8}}
=> ?
=> ? = 2
[4,3,6,1,7,2,8,5] => {{1,4},{2,3,6},{5,7,8}}
=> {{1,3,4},{2,7,8},{5,6}}
=> ?
=> ? = 3
[5,3,6,4,7,2,8,1] => {{1,5,7,8},{2,3,6},{4}}
=> {{1,3,5,6},{2,7,8},{4}}
=> ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[2,1] => [2]
=> [1,1]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => [3]
=> [2,1]
=> 2 = 1 + 1
[3,1,2] => [3]
=> [2,1]
=> 2 = 1 + 1
[3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [2,1,1]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[2,3,4,1] => [4]
=> [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [4]
=> [3,1]
=> 2 = 1 + 1
[2,4,3,1] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[3,1,2,4] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[3,1,4,2] => [4]
=> [3,1]
=> 2 = 1 + 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[3,4,1,2] => [2,2]
=> [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [4]
=> [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [4]
=> [3,1]
=> 2 = 1 + 1
[4,1,3,2] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [2,2]
=> 2 = 1 + 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> [3,1]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [2,1,1]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,5,2,4] => [4,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? = 1 + 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? = 1 + 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? = 1 + 1
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? = 2 + 1
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? = 1 + 1
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? = 1 + 1
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? = 1 + 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? = 1 + 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? = 1 + 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? = 1 + 1
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? = 1 + 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? = 1 + 1
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? = 1 + 1
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? = 1 + 1
[6,5,4,2,1,3,7,8] => ?
=> ?
=> ? = 1 + 1
[6,4,2,1,3,5,7,8] => ?
=> ?
=> ? = 1 + 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? = 1 + 1
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? = 1 + 1
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? = 2 + 1
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? = 1 + 1
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? = 1 + 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? = 2 + 1
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? = 1 + 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? = 1 + 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? = 1 + 1
[2,4,1,6,3,5,7,8] => ?
=> ?
=> ? = 1 + 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? = 1 + 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? = 1 + 1
[2,3,5,1,6,4,7,8] => ?
=> ?
=> ? = 1 + 1
[2,6,1,7,3,4,5,8] => ?
=> ?
=> ? = 1 + 1
[2,6,1,8,3,4,5,7] => ?
=> ?
=> ? = 1 + 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ? = 1 + 1
[3,4,6,1,2,7,5,8] => ?
=> ?
=> ? = 1 + 1
[3,4,6,7,1,2,8,5] => ?
=> ?
=> ? = 1 + 1
[3,5,6,1,7,2,4,8] => ?
=> ?
=> ? = 1 + 1
[3,6,7,1,2,4,8,5] => ?
=> ?
=> ? = 1 + 1
[3,1,6,7,2,4,5,8] => ?
=> ?
=> ? = 1 + 1
[3,1,6,2,4,7,5,8] => ?
=> ?
=> ? = 1 + 1
[4,5,1,2,7,3,6,8] => ?
=> ?
=> ? = 1 + 1
[4,5,1,2,8,3,6,7] => ?
=> ?
=> ? = 1 + 1
[5,1,7,2,3,8,4,6] => ?
=> ?
=> ? = 2 + 1
[5,1,8,2,3,4,6,7] => ?
=> ?
=> ? = 1 + 1
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ? = 1 + 1
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ? = 1 + 1
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ? = 1 + 1
[6,7,2,5,1,3,4,8] => ?
=> ?
=> ? = 1 + 1
[7,5,2,6,1,3,4,8] => ?
=> ?
=> ? = 1 + 1
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ? = 2 + 1
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ? = 1 + 1
[9,1,2,5,3,4,6,7,8] => ?
=> ?
=> ? = 1 + 1
Description
The length of the partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> []
=> 0
[1,2] => [1,1]
=> [2]
=> []
=> 0
[2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 1
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 1
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ?
=> ? = 1
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ?
=> ? = 1
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ?
=> ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ?
=> ? = 1
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ?
=> ? = 1
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ?
=> ? = 1
[6,5,4,2,1,3,7,8] => ?
=> ?
=> ?
=> ? = 1
[6,4,2,1,3,5,7,8] => ?
=> ?
=> ?
=> ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ?
=> ? = 1
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ?
=> ? = 1
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ?
=> ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ?
=> ? = 1
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ?
=> ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ?
=> ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ?
=> ? = 1
[2,4,1,6,3,5,7,8] => ?
=> ?
=> ?
=> ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ?
=> ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ?
=> ? = 1
[2,3,5,1,6,4,7,8] => ?
=> ?
=> ?
=> ? = 1
[2,6,1,7,3,4,5,8] => ?
=> ?
=> ?
=> ? = 1
[2,6,1,8,3,4,5,7] => ?
=> ?
=> ?
=> ? = 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 1
[3,4,6,1,2,7,5,8] => ?
=> ?
=> ?
=> ? = 1
[3,4,6,7,1,2,8,5] => ?
=> ?
=> ?
=> ? = 1
[3,5,6,1,7,2,4,8] => ?
=> ?
=> ?
=> ? = 1
[3,6,7,1,2,4,8,5] => ?
=> ?
=> ?
=> ? = 1
[3,1,6,7,2,4,5,8] => ?
=> ?
=> ?
=> ? = 1
[3,1,6,2,4,7,5,8] => ?
=> ?
=> ?
=> ? = 1
[4,5,1,2,7,3,6,8] => ?
=> ?
=> ?
=> ? = 1
[4,5,1,2,8,3,6,7] => ?
=> ?
=> ?
=> ? = 1
[5,1,7,2,3,8,4,6] => ?
=> ?
=> ?
=> ? = 2
[5,1,8,2,3,4,6,7] => ?
=> ?
=> ?
=> ? = 1
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ?
=> ? = 1
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 1
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ?
=> ? = 1
[6,7,2,5,1,3,4,8] => ?
=> ?
=> ?
=> ? = 1
[7,5,2,6,1,3,4,8] => ?
=> ?
=> ?
=> ? = 1
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ?
=> ? = 2
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ?
=> ? = 1
[9,1,2,5,3,4,6,7,8] => ?
=> ?
=> ?
=> ? = 1
Description
The largest part of an integer partition.
Matching statistic: St000665
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 1
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 1
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? => ? = 1
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? => ? = 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 1
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? => ? = 1
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? => ? = 1
[6,5,4,2,1,3,7,8] => ?
=> ?
=> ? => ? = 1
[6,4,2,1,3,5,7,8] => ?
=> ?
=> ? => ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? => ? = 1
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? => ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? => ? = 1
[2,4,1,6,3,5,7,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? => ? = 1
[2,3,5,1,6,4,7,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,7,3,4,5,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,8,3,4,5,7] => ?
=> ?
=> ? => ? = 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ? => ? = 1
[3,4,6,1,2,7,5,8] => ?
=> ?
=> ? => ? = 1
[3,4,6,7,1,2,8,5] => ?
=> ?
=> ? => ? = 1
[3,5,6,1,7,2,4,8] => ?
=> ?
=> ? => ? = 1
[3,6,7,1,2,4,8,5] => ?
=> ?
=> ? => ? = 1
[3,1,6,7,2,4,5,8] => ?
=> ?
=> ? => ? = 1
[3,1,6,2,4,7,5,8] => ?
=> ?
=> ? => ? = 1
[4,5,1,2,7,3,6,8] => ?
=> ?
=> ? => ? = 1
[4,5,1,2,8,3,6,7] => ?
=> ?
=> ? => ? = 1
[5,1,7,2,3,8,4,6] => ?
=> ?
=> ? => ? = 2
[5,1,8,2,3,4,6,7] => ?
=> ?
=> ? => ? = 1
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ? => ? = 1
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ? => ? = 1
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ? => ? = 1
[6,7,2,5,1,3,4,8] => ?
=> ?
=> ? => ? = 1
[7,5,2,6,1,3,4,8] => ?
=> ?
=> ? => ? = 1
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ? => ? = 2
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ? => ? = 1
[9,1,2,5,3,4,6,7,8] => ?
=> ?
=> ? => ? = 1
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St000834
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 1
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 1
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? => ? = 1
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? => ? = 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 1
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? => ? = 1
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? => ? = 1
[6,5,4,2,1,3,7,8] => ?
=> ?
=> ? => ? = 1
[6,4,2,1,3,5,7,8] => ?
=> ?
=> ? => ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? => ? = 1
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? => ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? => ? = 1
[2,4,1,6,3,5,7,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? => ? = 1
[2,3,5,1,6,4,7,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,7,3,4,5,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,8,3,4,5,7] => ?
=> ?
=> ? => ? = 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ? => ? = 1
[3,4,6,1,2,7,5,8] => ?
=> ?
=> ? => ? = 1
[3,4,6,7,1,2,8,5] => ?
=> ?
=> ? => ? = 1
[3,5,6,1,7,2,4,8] => ?
=> ?
=> ? => ? = 1
[3,6,7,1,2,4,8,5] => ?
=> ?
=> ? => ? = 1
[3,1,6,7,2,4,5,8] => ?
=> ?
=> ? => ? = 1
[3,1,6,2,4,7,5,8] => ?
=> ?
=> ? => ? = 1
[4,5,1,2,7,3,6,8] => ?
=> ?
=> ? => ? = 1
[4,5,1,2,8,3,6,7] => ?
=> ?
=> ? => ? = 1
[5,1,7,2,3,8,4,6] => ?
=> ?
=> ? => ? = 2
[5,1,8,2,3,4,6,7] => ?
=> ?
=> ? => ? = 1
[4,3,5,8,1,2,6,7] => ?
=> ?
=> ? => ? = 1
[3,2,6,8,1,4,5,7] => ?
=> ?
=> ? => ? = 1
[8,2,4,7,1,3,5,6] => ?
=> ?
=> ? => ? = 1
[6,7,2,5,1,3,4,8] => ?
=> ?
=> ? => ? = 1
[7,5,2,6,1,3,4,8] => ?
=> ?
=> ? => ? = 1
[6,3,2,8,1,4,5,7] => ?
=> ?
=> ? => ? = 2
[8,6,5,3,2,7,1,4] => ?
=> ?
=> ? => ? = 1
[9,1,2,5,3,4,6,7,8] => ?
=> ?
=> ? => ? = 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000319
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0 - 1
[1,2] => [1,1]
=> [2]
=> []
=> ? = 0 - 1
[2,1] => [2]
=> [1,1]
=> [1]
=> 0 = 1 - 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> ? = 0 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> ? = 0 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> ? = 0 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [6]
=> []
=> ? = 0 - 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? = 0 - 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,5,4,2,1,3,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,4,2,1,3,5,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [8]
=> []
=> ? = 0 - 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,4,1,6,3,5,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,3,5,1,6,4,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,6,1,7,3,4,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,6,1,8,3,4,5,7] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,4,6,1,2,7,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,4,6,7,1,2,8,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,5,6,1,7,2,4,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,6,7,1,2,4,8,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,1,6,7,2,4,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,1,6,2,4,7,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,5,1,2,7,3,6,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,5,1,2,8,3,6,7] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,1,7,2,3,8,4,6] => ?
=> ?
=> ?
=> ? = 2 - 1
[5,1,8,2,3,4,6,7] => ?
=> ?
=> ?
=> ? = 1 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0 - 1
[1,2] => [1,1]
=> [2]
=> []
=> ? = 0 - 1
[2,1] => [2]
=> [1,1]
=> [1]
=> 0 = 1 - 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> ? = 0 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> ? = 0 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> ? = 0 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [6]
=> []
=> ? = 0 - 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? = 0 - 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ?
=> ? = 1 - 1
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,5,4,2,1,3,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[6,4,2,1,3,5,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [8]
=> []
=> ? = 0 - 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ?
=> ? = 2 - 1
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,4,1,6,3,5,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,3,5,1,6,4,7,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,6,1,7,3,4,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,6,1,8,3,4,5,7] => ?
=> ?
=> ?
=> ? = 1 - 1
[2,3,4,6,1,7,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,4,6,1,2,7,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,4,6,7,1,2,8,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,5,6,1,7,2,4,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,6,7,1,2,4,8,5] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,1,6,7,2,4,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[3,1,6,2,4,7,5,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,5,1,2,7,3,6,8] => ?
=> ?
=> ?
=> ? = 1 - 1
[4,5,1,2,8,3,6,7] => ?
=> ?
=> ?
=> ? = 1 - 1
[5,1,7,2,3,8,4,6] => ?
=> ?
=> ?
=> ? = 2 - 1
[5,1,8,2,3,4,6,7] => ?
=> ?
=> ?
=> ? = 1 - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000245The number of ascents of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle 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. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000325The width of the tree associated to 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice.
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