- FindStat

Your data matches 456 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001809
(load all 19 compositions to match this statistic)

Mapping

St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => 6 = 7 - 1
[1,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => 4 = 5 - 1

Description

The index of the step at the first peak of maximal height in a Dyck path.

Matching statistic: St000738

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [(1,2)]
 => [2,1] => [[1],[2]]
 => 2
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [[1,4],[2],[3]]
 => 3
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [[1,3,5],[2,4,6]]
 => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [[1,3,6],[2,4],[5]]
 => 5
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [[1,4,5],[2,6],[3]]
 => 3
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [[1,4,6],[2,5],[3]]
 => 3
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [[1,5,6],[2],[3],[4]]
 => 4
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [[1,3,5,7],[2,4,6,8]]
 => 2
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [[1,3,5,8],[2,4,6],[7]]
 => 7
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [[1,3,6,7],[2,4,8],[5]]
 => 5
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [[1,3,6,8],[2,4,7],[5]]
 => 5
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => [[1,3,7,8],[2,4],[5],[6]]
 => 6
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [[1,4,5,7],[2,6,8],[3]]
 => 3
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => [[1,4,5,8],[2,6],[3,7]]
 => 3
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [[1,4,6,7],[2,5,8],[3]]
 => 3
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [[1,4,6,8],[2,5,7],[3]]
 => 3
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [[1,4,7,8],[2,5],[3],[6]]
 => 6
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [[1,5,6,7],[2,8],[3],[4]]
 => 4
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [[1,5,6,8],[2,7],[3],[4]]
 => 4
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [[1,5,7,8],[2,6],[3],[4]]
 => 4
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [[1,6,7,8],[2],[3],[4],[5]]
 => 5

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St000238

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000238: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,2] => [1,2] => 0 = 2 - 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [1,3,2] => 1 = 3 - 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0 = 2 - 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => 3 = 5 - 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [1,3,2,4] => 1 = 3 - 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,4,2,3,5] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,1,4,5,2] => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [1,2,4,5,3] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [1,3,2,4,5] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,1,5,3,4] => 4 = 6 - 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => [2,1,4,3,5] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [3,1,5,2,4] => 5 = 7 - 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,3,1,5,4] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [2,1,3,5,4] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2

Description

The number of indices that are not small weak excedances. A small weak excedance is an index

$i$ such that

$\pi_i \in \{i,i+1\}$ .

Matching statistic: St000242

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,2] => [1,2] => 0 = 2 - 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => [1,3,2] => 1 = 3 - 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0 = 2 - 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => 3 = 5 - 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [1,3,2,4] => 1 = 3 - 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,4,2,3,5] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,1,4,5,2] => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [1,2,4,5,3] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [1,3,2,4,5] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,1,5,3,4] => 4 = 6 - 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => [2,1,4,3,5] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [3,1,5,2,4] => 5 = 7 - 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,3,1,5,4] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [2,1,3,5,4] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2

Description

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index

$i$ such that

$\pi_i \in \{ i,i+1 \}$ considered cyclically.

Matching statistic: St000472

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1] => ? = 2 - 2
[1,0,1,0]
 => [2,1] => [2,1] => [1,2] => 1 = 3 - 2
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 0 = 2 - 2
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => [1,2,3] => 3 = 5 - 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => [3,2,1] => 0 = 2 - 2
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => [1,3,2] => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [2,3,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => [2,1,3,4] => 4 = 6 - 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,4,1,2] => [4,1,3,2] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => [4,2,3,1] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 4 = 6 - 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 5 = 7 - 2

Description

The sum of the ascent bottoms of a permutation.

Matching statistic: St001782
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001782: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [[1]]
 => [[1]]
 => ([],1)
 => 2
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => 2
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => 3
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => 2
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => 3
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => 3
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => 4
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ([],1)
 => 2
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ([(0,1)],2)
 => 3
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ([(0,1)],2)
 => 3
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => ([(0,2),(2,1)],3)
 => 4
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ([(0,1)],2)
 => 3
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => ([(0,2),(2,1)],3)
 => 4
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {3,6,7}
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {3,6,7}
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ? ∊ {3,6,7}

Description

The order of rowmotion on the set of order ideals of a poset.

Matching statistic: St000528
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000528: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [[1]]
 => [[1]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => 1 = 2 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {3,6,7} - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5 = 6 - 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {3,6,7} - 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ? ∊ {3,6,7} - 1

Description

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

Matching statistic: St000912
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000912: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [[1]]
 => [[1]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => 1 = 2 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {3,6,7} - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5 = 6 - 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {3,6,7} - 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ? ∊ {3,6,7} - 1

Description

The number of maximal antichains in a poset.

Matching statistic: St001636
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001636: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [[1]]
 => [[1]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => 1 = 2 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {3,6,7} - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5 = 6 - 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {3,6,7} - 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ? ∊ {3,6,7} - 1

Description

The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.

Matching statistic: St000080
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000080: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [[1]]
 => [[1]]
 => ([],1)
 => 0 = 2 - 2
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,1],[2]]
 => ([],1)
 => 0 = 2 - 2
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ([],1)
 => 0 = 2 - 2
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[1,1,2],[2,3],[3]]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ([],1)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,3],[3,4],[4]]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,3],[3,3],[4]]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,3],[3,4],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,2],[2,2,4],[3,4],[4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? ∊ {3,6,7} - 2
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,1,1,3],[2,2,3],[3,4],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4 = 6 - 2
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,3],[2,2,4],[3,4],[4]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {3,6,7} - 2
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ? ∊ {3,6,7} - 2

Description

The rank of the poset.

The following 446 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000141The maximum drop size of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000356The number of occurrences of the pattern 13-2. St000454The largest eigenvalue of a graph if it is integral. St000670The reversal length of a permutation. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000189The number of elements in the poset. St000907The number of maximal antichains of minimal length in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001645The pebbling number of a connected graph. St001717The largest size of an interval in a poset. St001894The depth of a signed permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St000490The intertwining number of a set partition. St000499The rcb statistic of a set partition. St001861The number of Bruhat lower covers of a permutation. St000441The number of successions of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000662The staircase size of the code of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000366The number of double descents of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St000028The number of stack-sorts needed to sort a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000054The first entry of the permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000451The length of the longest pattern of the form k 1 2. St000456The monochromatic index of a connected graph. St000691The number of changes of a binary word. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001090The number of pop-stack-sorts needed to sort a permutation. St001330The hat guessing number of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000359The number of occurrences of the pattern 23-1. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001415The length of the longest palindromic prefix of a binary word. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000173The segment statistic of a semistandard tableau. St000360The number of occurrences of the pattern 32-1. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. 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. St001388The number of non-attacking neighbors of a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. 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. 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. St001727The number of invisible inversions of a permutation. St001843The Z-index of a set partition. St000325The width of the tree associated to a permutation. St000422The energy of a graph, if it is integral. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000638The number of up-down runs of a permutation. St000675The number of centered multitunnels of a Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001116The game chromatic number of a graph. St001315The dissociation number of a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001618The cardinality of the Frattini sublattice of a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000021The number of descents of a permutation. St000058The order of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000570The Edelman-Greene number of a permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000829The Ulam distance of a permutation to the identity permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001246The maximal difference between two consecutive entries of a permutation. St001270The bandwidth of a graph. St001282The number of graphs with the same chromatic polynomial. St001405The number of bonds in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001760The number of prefix or suffix reversals needed to sort a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001821The sorting index of a signed permutation. St001874Lusztig's a-function for the symmetric group. St000006The dinv of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000091The descent variation of a composition. St000136The dinv of a parking function. St000174The flush statistic of a semistandard tableau. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000516The number of stretching pairs of a permutation. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000809The reduced reflection length of the permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001209The pmaj statistic of a parking function. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001403The number of vertical separators in a permutation. St001535The number of cyclic alignments of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001841The number of inversions of a set partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000438The position of the last up step in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001623The number of doubly irreducible elements of a lattice. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001644The dimension of a graph. St000850The number of 1/2-balanced pairs in a poset. St001875The number of simple modules with projective dimension at most 1. St000100The number of linear extensions of a poset. St001812The biclique partition number of a graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000264The girth of a graph, which is not a tree. St000893The number of distinct diagonal sums of an alternating sign matrix. St001060The distinguishing index of a graph. St000741The Colin de Verdière graph invariant. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000527The width of the poset. St000617The number of global maxima of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000909The number of maximal chains of maximal size in a poset. St001820The size of the image of the pop stack sorting operator. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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

$K[x]/(x^n)$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000254The nesting number of a set partition. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000492The rob statistic of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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. St000654The first descent of a permutation. St000739The first entry in the last row of a semistandard tableau. St000824The sum of the number of descents and the number of recoils of a permutation. St000842The breadth of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001114The number of odd descents of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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

$eA$ . St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. 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. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001637The number of (upper) dissectors of a poset. St001667The maximal size of a pair of weak twins for a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000007The number of saliances of the permutation. St000023The number of inner peaks of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000062The length of the longest increasing subsequence of the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000094The depth of an ordered tree. St000239The number of small weak excedances. St000253The crossing number of a set partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000315The number of isolated vertices of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000383The last part of an integer composition. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000542The number of left-to-right-minima of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000779The tier of a permutation. St000834The number of right outer peaks of a permutation. St000839The largest opener of a set partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000988The orbit size of a permutation under Foata's bijection. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001115The number of even descents of a permutation. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. 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. 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. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001487The number of inner corners of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001728The number of invisible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001823The Stasinski-Voll length of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001862The number of crossings of a signed permutation. 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). St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000232The number of crossings of a set partition. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000562The number of internal points of a set partition. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000632The jump number of the poset. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000871The number of very big ascents of a permutation. St000884The number of isolated descents of a permutation. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001153The number of blocks with even minimum in a set partition. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001186Number of simple modules with grade at least 3 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001377The major index minus the number of inversions of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001684The reduced word complexity of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001911A descent variant minus the number of inversions. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset.