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Your data matches 509 different statistics following compositions of up to 3 maps.
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Matching statistic: St000757
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Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000757: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
Matching statistic: St000765
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(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The number of weak records in an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
Matching statistic: St000899
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(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000900
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(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000902
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(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000904
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(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001675
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(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001675: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001675: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,1] => [2] => 1 = 2 - 1
[2] => [1] => 1 = 2 - 1
[1,1,1] => [3] => 1 = 2 - 1
[1,2] => [1,1] => 2 = 3 - 1
[2,1] => [1,1] => 2 = 3 - 1
[3] => [1] => 1 = 2 - 1
[1,1,1,1] => [4] => 1 = 2 - 1
[1,2,1] => [1,1,1] => 3 = 4 - 1
[1,3] => [1,1] => 2 = 3 - 1
[2,2] => [2] => 1 = 2 - 1
[3,1] => [1,1] => 2 = 3 - 1
[4] => [1] => 1 = 2 - 1
[1,1,1,1,1] => [5] => 1 = 2 - 1
[1,3,1] => [1,1,1] => 3 = 4 - 1
[1,4] => [1,1] => 2 = 3 - 1
[2,1,2] => [1,1,1] => 3 = 4 - 1
[2,3] => [1,1] => 2 = 3 - 1
[3,2] => [1,1] => 2 = 3 - 1
[4,1] => [1,1] => 2 = 3 - 1
[5] => [1] => 1 = 2 - 1
[1,1,1,1,1,1] => [6] => 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,2,3] => [1,1,1] => 3 = 4 - 1
[1,3,2] => [1,1,1] => 3 = 4 - 1
[1,4,1] => [1,1,1] => 3 = 4 - 1
[1,5] => [1,1] => 2 = 3 - 1
[2,1,2,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,3] => [1,1,1] => 3 = 4 - 1
[2,2,2] => [3] => 1 = 2 - 1
[2,3,1] => [1,1,1] => 3 = 4 - 1
[2,4] => [1,1] => 2 = 3 - 1
[3,1,2] => [1,1,1] => 3 = 4 - 1
[3,2,1] => [1,1,1] => 3 = 4 - 1
[3,3] => [2] => 1 = 2 - 1
[4,2] => [1,1] => 2 = 3 - 1
[5,1] => [1,1] => 2 = 3 - 1
[6] => [1] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => 5 = 6 - 1
[1,2,1,3] => [1,1,1,1] => 4 = 5 - 1
[1,2,3,1] => [1,1,1,1] => 4 = 5 - 1
[1,2,4] => [1,1,1] => 3 = 4 - 1
[1,3,1,2] => [1,1,1,1] => 4 = 5 - 1
[1,3,2,1] => [1,1,1,1] => 4 = 5 - 1
[1,4,2] => [1,1,1] => 3 = 4 - 1
[1,5,1] => [1,1,1] => 3 = 4 - 1
[1,6] => [1,1] => 2 = 3 - 1
[2,1,3,1] => [1,1,1,1] => 4 = 5 - 1
[2,1,4] => [1,1,1] => 3 = 4 - 1
Description
The number of parts equal to the part in the reversed composition.
Matching statistic: St001777
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001777: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,1] => [2] => 0 = 2 - 2
[2] => [1] => 0 = 2 - 2
[1,1,1] => [3] => 0 = 2 - 2
[1,2] => [1,1] => 1 = 3 - 2
[2,1] => [1,1] => 1 = 3 - 2
[3] => [1] => 0 = 2 - 2
[1,1,1,1] => [4] => 0 = 2 - 2
[1,2,1] => [1,1,1] => 2 = 4 - 2
[1,3] => [1,1] => 1 = 3 - 2
[2,2] => [2] => 0 = 2 - 2
[3,1] => [1,1] => 1 = 3 - 2
[4] => [1] => 0 = 2 - 2
[1,1,1,1,1] => [5] => 0 = 2 - 2
[1,3,1] => [1,1,1] => 2 = 4 - 2
[1,4] => [1,1] => 1 = 3 - 2
[2,1,2] => [1,1,1] => 2 = 4 - 2
[2,3] => [1,1] => 1 = 3 - 2
[3,2] => [1,1] => 1 = 3 - 2
[4,1] => [1,1] => 1 = 3 - 2
[5] => [1] => 0 = 2 - 2
[1,1,1,1,1,1] => [6] => 0 = 2 - 2
[1,2,1,2] => [1,1,1,1] => 3 = 5 - 2
[1,2,3] => [1,1,1] => 2 = 4 - 2
[1,3,2] => [1,1,1] => 2 = 4 - 2
[1,4,1] => [1,1,1] => 2 = 4 - 2
[1,5] => [1,1] => 1 = 3 - 2
[2,1,2,1] => [1,1,1,1] => 3 = 5 - 2
[2,1,3] => [1,1,1] => 2 = 4 - 2
[2,2,2] => [3] => 0 = 2 - 2
[2,3,1] => [1,1,1] => 2 = 4 - 2
[2,4] => [1,1] => 1 = 3 - 2
[3,1,2] => [1,1,1] => 2 = 4 - 2
[3,2,1] => [1,1,1] => 2 = 4 - 2
[3,3] => [2] => 0 = 2 - 2
[4,2] => [1,1] => 1 = 3 - 2
[5,1] => [1,1] => 1 = 3 - 2
[6] => [1] => 0 = 2 - 2
[1,1,1,1,1,1,1] => [7] => 0 = 2 - 2
[1,2,1,2,1] => [1,1,1,1,1] => 4 = 6 - 2
[1,2,1,3] => [1,1,1,1] => 3 = 5 - 2
[1,2,3,1] => [1,1,1,1] => 3 = 5 - 2
[1,2,4] => [1,1,1] => 2 = 4 - 2
[1,3,1,2] => [1,1,1,1] => 3 = 5 - 2
[1,3,2,1] => [1,1,1,1] => 3 = 5 - 2
[1,4,2] => [1,1,1] => 2 = 4 - 2
[1,5,1] => [1,1,1] => 2 = 4 - 2
[1,6] => [1,1] => 1 = 3 - 2
[2,1,3,1] => [1,1,1,1] => 3 = 5 - 2
[2,1,4] => [1,1,1] => 2 = 4 - 2
Description
The number of weak descents in an integer composition.
A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.
Matching statistic: St000969
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,1] => [2] => [1,1,0,0]
=> 2
[2] => [1] => [1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 2
[1,2] => [1,1] => [1,0,1,0]
=> 3
[2,1] => [1,1] => [1,0,1,0]
=> 3
[3] => [1] => [1,0]
=> 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3] => [1,1] => [1,0,1,0]
=> 3
[2,2] => [2] => [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> 3
[4] => [1] => [1,0]
=> 2
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,4] => [1,1] => [1,0,1,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3] => [1,1] => [1,0,1,0]
=> 3
[3,2] => [1,1] => [1,0,1,0]
=> 3
[4,1] => [1,1] => [1,0,1,0]
=> 3
[5] => [1] => [1,0]
=> 2
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,5] => [1,1] => [1,0,1,0]
=> 3
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,4] => [1,1] => [1,0,1,0]
=> 3
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,3] => [2] => [1,1,0,0]
=> 2
[4,2] => [1,1] => [1,0,1,0]
=> 3
[5,1] => [1,1] => [1,0,1,0]
=> 3
[6] => [1] => [1,0]
=> 2
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,6] => [1,1] => [1,0,1,0]
=> 3
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 4
Description
We 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 $c_{n-1}$. Then we calculate the global dimension of that CNakayama algebra.
Matching statistic: St001028
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,1] => [2] => [1,1,0,0]
=> 2
[2] => [1] => [1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 2
[1,2] => [1,1] => [1,0,1,0]
=> 3
[2,1] => [1,1] => [1,0,1,0]
=> 3
[3] => [1] => [1,0]
=> 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3] => [1,1] => [1,0,1,0]
=> 3
[2,2] => [2] => [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> 3
[4] => [1] => [1,0]
=> 2
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,4] => [1,1] => [1,0,1,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3] => [1,1] => [1,0,1,0]
=> 3
[3,2] => [1,1] => [1,0,1,0]
=> 3
[4,1] => [1,1] => [1,0,1,0]
=> 3
[5] => [1] => [1,0]
=> 2
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,5] => [1,1] => [1,0,1,0]
=> 3
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,4] => [1,1] => [1,0,1,0]
=> 3
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,3] => [2] => [1,1,0,0]
=> 2
[4,2] => [1,1] => [1,0,1,0]
=> 3
[5,1] => [1,1] => [1,0,1,0]
=> 3
[6] => [1] => [1,0]
=> 2
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,6] => [1,1] => [1,0,1,0]
=> 3
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 4
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
The following 499 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000160The multiplicity of the smallest part of a partition. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000657The smallest part of an integer composition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000722The number of different neighbourhoods in a graph. St000808The number of up steps of the associated bargraph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to 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 Dyck path as follows:
St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. St001933The largest multiplicity of a part in an integer partition. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St000272The treewidth of a graph. St000295The length of the border of a binary word. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000054The first entry of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000294The number of distinct factors of a binary word. St000300The number of independent sets of vertices of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000518The number of distinct subsequences in a binary word. St000532The total number of rook placements on a Ferrers board. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001330The hat guessing number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001814The number of partitions interlacing the given partition. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000258The burning number of a graph. St000273The domination number of a graph. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000636The hull number of a graph. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000703The number of deficiencies of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000839The largest opener of a set partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000926The clique-coclique number of a graph. St000982The length of the longest constant subword. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. 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. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001058The breadth of the ordered tree. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001485The modular major index of a binary word. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001614The cyclic permutation representation number of a skew partition. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001910The height of the middle non-run of a Dyck path. St000008The major index of the composition. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000145The Dyson rank of a partition. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000546The number of global descents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000867The sum of the hook lengths in the first row of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001127The sum of the squares of the parts of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001340The cardinality of a minimal non-edge isolating set of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001484The number of singletons of an integer partition. St001759The Rajchgot index of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000308The height of the tree associated to a permutation. St000740The last entry of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000376The bounce deficit of a Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000678The number of up steps after the last double rise of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000729The minimal arc length of a set partition. St000770The major index of an integer partition when read from bottom to top. St000925The number of topologically connected components of a set partition. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000654The first descent of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000989The number of final rises of a permutation. St000806The semiperimeter of the associated bargraph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000438The position of the last up step in a Dyck path. St000477The weight of a partition according to Alladi. St000675The number of centered multitunnels of a Dyck path. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000931The number of occurrences of the pattern UUU in a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000306The bounce count of a Dyck path. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000301The number of facets of the stable set polytope of a graph. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000087The number of induced subgraphs. St000117The number of centered tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000822The Hadwiger number of the graph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective 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. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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 $g$. 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. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001481The minimal height of a peak of a Dyck path. St001828The Euler characteristic of a graph. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000446The disorder of a permutation. St000741The Colin de Verdière graph invariant. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001391The disjunction number of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000015The number of peaks of a Dyck path. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001530The depth of a Dyck path. St000331The number of upper interactions of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000164The number of short pairs. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000390The number of runs of ones in a binary word. St000638The number of up-down runs of a permutation. St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000004The major index of a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St001172The number of 1-rises at odd height of a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000840The number of closers smaller than the largest opener in a perfect matching. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000990The first ascent of a permutation. St000354The number of recoils 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. St000653The last descent of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St000061The number of nodes on the left branch of a binary tree. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. 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. St001834The number of non-isomorphic minors of a graph. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000159The number of distinct parts of the integer partition. St001812The biclique partition number of a graph. St001458The rank of the adjacency matrix of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000305The inverse major index of a permutation. St000702The number of weak deficiencies of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000225Difference between largest and smallest parts in a partition. St001280The number of parts of an integer partition that are at least two. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001948The number of augmented double ascents of a permutation. St001432The order dimension of the partition. St000474Dyson's crank of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001712The number of natural descents of a standard Young tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $eA$. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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