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Your data matches 591 different statistics following compositions of up to 3 maps.
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Matching statistic: St001252
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St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 2
[3,1]
=> 0
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 2
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
Description
Half the sum of the even parts of a partition.
Matching statistic: St001414
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(load all 7 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00280: Binary words —path rowmotion⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => 0
[2]
=> 100 => 011 => 0
[1,1]
=> 110 => 111 => 1
[3]
=> 1000 => 0011 => 0
[2,1]
=> 1010 => 1101 => 0
[1,1,1]
=> 1110 => 1111 => 1
[4]
=> 10000 => 00011 => 1
[3,1]
=> 10010 => 01101 => 0
[2,2]
=> 1100 => 0111 => 0
[2,1,1]
=> 10110 => 11011 => 2
[1,1,1,1]
=> 11110 => 11111 => 2
[5]
=> 100000 => 000011 => 1
[4,1]
=> 100010 => 001101 => 0
[3,2]
=> 10100 => 11001 => 0
[3,1,1]
=> 100110 => 011011 => 0
[2,2,1]
=> 11010 => 11101 => 1
[2,1,1,1]
=> 101110 => 110111 => 2
[1,1,1,1,1]
=> 111110 => 111111 => 2
Description
Half the length of the longest odd length palindromic prefix of a binary word.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
Matching statistic: St001584
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(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
Description
The area statistic between a Dyck path and its bounce path.
The bounce path [[Mp00099]] is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.
Matching statistic: St001801
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(load all 4 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
Description
Half the number of preimage-image pairs of different parity in a permutation.
Matching statistic: St000909
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Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> ([],1)
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> ([],2)
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> ([],3)
=> 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 1 = 0 + 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000143
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(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 0
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000228
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(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> 0
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> 0
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000355
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(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [2,3,4,1,5] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,3,4,5,2] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [2,3,4,5,1,6] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,4,2,1,5] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,4,5,3,2] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,3,4,5,6,2] => 0
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000359
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> [1,4,3,2,6,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [2,4,3,6,5,1] => 2
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000384
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> 0
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> 0
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
The following 581 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000459The hook length of the base cell of a partition. St000463The number of admissible inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000565The major index of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000784The maximum of the length and the largest part of the integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001176The size of a partition minus its first part. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001377The major index minus the number of inversions of a permutation. St001388The number of non-attacking neighbors of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001727The number of invisible inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000025The number of initial rises of a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000120The number of left tunnels of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000532The total number of rook placements on a Ferrers board. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001400The total number of Littlewood-Richardson tableaux of given shape. St001480The number of simple summands of the module J^2/J^3. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St000058The order of a permutation. St000439The position of the first down step of a Dyck path. St000485The length of the longest cycle of a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001530The depth of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000516The number of stretching pairs of a permutation. St000663The number of right floats of a permutation. St000989The number of final rises of a permutation. St001535The number of cyclic alignments of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000383The last part of an integer composition. St000740The last entry of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000644The number of graphs with given frequency partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000013The height of a Dyck path. St000053The number of valleys of the Dyck path. St000117The number of centered tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000154The sum of the descent bottoms of a permutation. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000378The diagonal inversion number of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000877The depth of the binary word interpreted as a path. St000895The number of ones on the main diagonal of an alternating sign matrix. St000921The number of internal inversions of a binary word. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001760The number of prefix or suffix reversals needed to sort a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001910The height of the middle non-run of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001956The comajor index for set-valued two-row standard Young tableaux. St001964The interval resolution global dimension of a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St000461The rix statistic of a permutation. St000761The number of ascents in an integer composition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St000765The number of weak records in an integer composition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001737The number of descents of type 2 in a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000638The number of up-down runs of a permutation. St000478Another weight of a partition according to Alladi. St001487The number of inner corners of a skew partition. St001488The number of corners of a skew partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000137The Grundy value of an integer partition. St000216The absolute length of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000259The diameter of a connected graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000353The number of inner valleys of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000462The major index minus the number of excedences of a permutation. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000491The number of inversions of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000538The number of even inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000572The dimension exponent of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000831The number of indices that are either descents or recoils. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000961The shifted major index of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001524The degree of symmetry of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001721The degree of a binary word. St001731The factorization defect of a permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000352The Elizalde-Pak rank of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000054The first entry of the permutation. St000944The 3-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000010The length of the partition. St000012The area of a Dyck path. St000142The number of even parts of a partition. St000144The pyramid weight of the Dyck path. St000145The Dyson rank of a partition. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000185The weighted size of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000260The radius of a connected graph. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000345The number of refinements of a partition. St000377The dinv defect of an integer partition. St000443The number of long tunnels of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000649The number of 3-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St000995The largest even part of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001127The sum of the squares of the parts of a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001172The number of 1-rises at odd height of a Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001389The number of partitions of the same length below the given integer partition. St001423The number of distinct cubes in a binary word. St001481The minimal height of a peak of a Dyck path. St001484The number of singletons of an integer partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001541The Gini index of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001657The number of twos in an integer partition. St001732The number of peaks visible from the left. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001961The sum of the greatest common divisors of all pairs of parts. St000492The rob statistic of a set partition. St000654The first descent of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000942The number of critical left to right maxima of the parking functions. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000542The number of left-to-right-minima of a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000937The number of positive values of the symmetric group character corresponding to the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000741The Colin de Verdière graph invariant. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000356The number of occurrences of the pattern 13-2. St000834The number of right outer peaks of a permutation. St000007The number of saliances of the permutation. 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. St000045The number of linear extensions of a binary tree. St000091The descent variation of a composition. St000365The number of double ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000562The number of internal points of a set partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000770The major index of an integer partition when read from bottom to top. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000284The Plancherel distribution on integer partitions. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000477The weight of a partition according to Alladi. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000928The sum of the coefficients of the character polynomial of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000997The even-odd crank of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001808The box weight or horizontal decoration of a Dyck path. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001943The sum of the squares of the hook lengths of an integer partition. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000717The number of ordinal summands of a poset.
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