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Your data matches 526 different statistics following compositions of up to 3 maps.
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Matching statistic: St000768
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 0
[2,1] => 0
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 1
[1,3] => 0
[2,1,1] => 0
[2,2] => 0
[3,1] => 0
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 0
[2,1,1,1] => 0
[5] => 0
[1,1,1,1,1,1] => 0
[2,1,1,2] => 0
[6] => 0
[1,1,1,1,1,1,1] => 0
[7] => 0
[1,1,1,1,1,1,1,1] => 0
[8] => 0
[9] => 0
Description
The number of peaks in an integer composition.
A peak is an ascent followed by a descent, i.e., a subsequence ci−1cici+1 with ci>max.
Matching statistic: St000404
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [2,1] => 0
[2] => [1,1,0,0]
=> [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => 0
Description
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.
A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].
Matching statistic: St000405
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [1,2] => 0
[2] => [1,1,0,0]
=> [2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => 0
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by S_n(1324)), but it is shown in [1], improving bounds in [2] and [3] that
\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.
Matching statistic: St000408
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [2,1] => 0
[2] => [1,1,0,0]
=> [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => 0
Description
The number of occurrences of the pattern 4231 in a permutation.
It is a necessary condition that a permutation \pi avoids this pattern for the Schubert variety associated to \pi to be smooth [2].
Matching statistic: St000440
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [2,1] => 0
[2] => [1,1,0,0]
=> [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => 0
Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.
There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
Matching statistic: St000761
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,1] => [1,1] => [2] => 0
[2] => [2] => [1] => 0
[1,1,1] => [1,1,1] => [3] => 0
[1,2] => [1,2] => [1,1] => 0
[2,1] => [2,1] => [1,1] => 0
[3] => [3] => [1] => 0
[1,1,1,1] => [1,1,1,1] => [4] => 0
[1,1,2] => [1,1,2] => [2,1] => 0
[1,2,1] => [2,1,1] => [1,2] => 1
[1,3] => [1,3] => [1,1] => 0
[2,1,1] => [1,2,1] => [1,1,1] => 0
[2,2] => [2,2] => [2] => 0
[3,1] => [3,1] => [1,1] => 0
[4] => [4] => [1] => 0
[1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[1,1,1,2] => [1,1,1,2] => [3,1] => 0
[2,1,1,1] => [1,1,2,1] => [2,1,1] => 0
[5] => [5] => [1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[2,1,1,2] => [1,2,1,2] => [1,1,1,1] => 0
[6] => [6] => [1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7] => [7] => [1] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8] => [8] => [1] => 0
[9] => [9] => [1] => 0
Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position i if a_i < a_{i+1}.
Matching statistic: St000766
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,1] => [1,1] => [2] => 0
[2] => [2] => [1] => 0
[1,1,1] => [1,1,1] => [3] => 0
[1,2] => [2,1] => [1,1] => 0
[2,1] => [1,2] => [1,1] => 0
[3] => [3] => [1] => 0
[1,1,1,1] => [1,1,1,1] => [4] => 0
[1,1,2] => [2,1,1] => [1,2] => 0
[1,2,1] => [1,1,2] => [2,1] => 1
[1,3] => [3,1] => [1,1] => 0
[2,1,1] => [1,2,1] => [1,1,1] => 0
[2,2] => [2,2] => [2] => 0
[3,1] => [1,3] => [1,1] => 0
[4] => [4] => [1] => 0
[1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[1,1,1,2] => [2,1,1,1] => [1,3] => 0
[2,1,1,1] => [1,2,1,1] => [1,1,2] => 0
[5] => [5] => [1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[2,1,1,2] => [2,2,1,1] => [2,2] => 0
[6] => [6] => [1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7] => [7] => [1] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8] => [8] => [1] => 0
[9] => [9] => [1] => 0
Description
The number of inversions of an integer composition.
This is the number of pairs (i,j) such that i < j and c_i > c_j.
Matching statistic: St000769
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,1] => [1,1] => [2] => 0
[2] => [2] => [1] => 0
[1,1,1] => [1,1,1] => [3] => 0
[1,2] => [2,1] => [1,1] => 0
[2,1] => [1,2] => [1,1] => 0
[3] => [3] => [1] => 0
[1,1,1,1] => [1,1,1,1] => [4] => 0
[1,1,2] => [2,1,1] => [1,2] => 0
[1,2,1] => [1,1,2] => [2,1] => 1
[1,3] => [3,1] => [1,1] => 0
[2,1,1] => [1,2,1] => [1,1,1] => 0
[2,2] => [2,2] => [2] => 0
[3,1] => [1,3] => [1,1] => 0
[4] => [4] => [1] => 0
[1,1,1,1,1] => [1,1,1,1,1] => [5] => 0
[1,1,1,2] => [2,1,1,1] => [1,3] => 0
[2,1,1,1] => [1,2,1,1] => [1,1,2] => 0
[5] => [5] => [1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 0
[2,1,1,2] => [2,2,1,1] => [2,2] => 0
[6] => [6] => [1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 0
[7] => [7] => [1] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 0
[8] => [8] => [1] => 0
[9] => [9] => [1] => 0
Description
The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see [[St000008]].
Matching statistic: St001137
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,1] => [2] => [1,1,0,0]
=> 0
[2] => [1] => [1,0]
=> 0
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,2] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 0
[3] => [1] => [1,0]
=> 0
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3] => [1,1] => [1,0,1,0]
=> 0
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,2] => [2] => [1,1,0,0]
=> 0
[3,1] => [1,1] => [1,0,1,0]
=> 0
[4] => [1] => [1,0]
=> 0
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[5] => [1] => [1,0]
=> 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[6] => [1] => [1,0]
=> 0
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[7] => [1] => [1,0]
=> 0
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 0
[8] => [1] => [1,0]
=> 0
[9] => [1] => [1,0]
=> 0
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St001175
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,1] => [2] => [2]
=> 0
[2] => [1,1] => [1,1]
=> 0
[1,1,1] => [3] => [3]
=> 0
[1,2] => [2,1] => [2,1]
=> 0
[2,1] => [1,2] => [2,1]
=> 0
[3] => [1,1,1] => [1,1,1]
=> 0
[1,1,1,1] => [4] => [4]
=> 0
[1,1,2] => [3,1] => [3,1]
=> 0
[1,2,1] => [2,2] => [2,2]
=> 1
[1,3] => [2,1,1] => [2,1,1]
=> 0
[2,1,1] => [1,3] => [3,1]
=> 0
[2,2] => [1,2,1] => [2,1,1]
=> 0
[3,1] => [1,1,2] => [2,1,1]
=> 0
[4] => [1,1,1,1] => [1,1,1,1]
=> 0
[1,1,1,1,1] => [5] => [5]
=> 0
[1,1,1,2] => [4,1] => [4,1]
=> 0
[2,1,1,1] => [1,4] => [4,1]
=> 0
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,1,1,1,1,1] => [6] => [6]
=> 0
[2,1,1,2] => [1,4,1] => [4,1,1]
=> 0
[6] => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1] => [7] => [7]
=> 0
[7] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1] => [8] => [8]
=> 0
[8] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 0
[9] => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> 0
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
The following 516 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000183The side length of the Durfee square of an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000057The Shynar inversion number of a standard tableau. St000091The descent variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents 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. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000731The number of double exceedences of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000871The number of very big ascents of a permutation. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000944The 3-degree of an integer partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001172The number of 1-rises at odd height of a Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001394The genus of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. St000001The number of reduced words for a permutation. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000079The number of alternating sign matrices for a given Dyck path. St000182The number of permutations whose cycle type is the given integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000655The length of the minimal rise of a Dyck path. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000816The number of standard composition tableaux of the composition. St000862The number of parts of the shifted shape of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000905The number of different multiplicities of parts of an integer composition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001917The order of toric promotion on the set of labellings of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000842The breadth of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001797The number of overfull subgraphs of a graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000017The number of inversions of a standard tableau. St000052The number of valleys of a Dyck path not on the x-axis. St000233The number of nestings of a set partition. St000367The number of simsun double descents of a permutation. St000663The number of right floats of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001732The number of peaks visible from the left. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000039The number of crossings of a permutation. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by 4. St000365The number of double ascents of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001196The global dimension of A minus the global dimension of eAe for the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000552The number of cut vertices of a graph. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001689The number of celebrities in a graph. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000562The number of internal points of a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (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. St000599The number of occurrences of the pattern {{1},{2,3}} such that (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. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000649The number of 3-excedences of a permutation. St000664The number of right ropes of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001411The number of patterns 321 or 3412 in a permutation. St001521Half the total irregularity of a graph. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001596The number of two-by-two squares inside a skew partition. St001638The book thickness of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000388The number of orbits of vertices of a graph under automorphisms. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000968We 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}. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. 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. St001352The number of internal nodes in the modular decomposition of a graph. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000407The number of occurrences of the pattern 2143 in a permutation. St001323The independence gap of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000406The number of occurrences of the pattern 3241 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000658The number of rises of length 2 of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000872The number of very big descents of a permutation. St001377The major index minus the number of inversions of a permutation. St001513The number of nested exceedences of a permutation. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001728The number of invisible descents of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001734The lettericity of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000252The number of nodes of degree 3 of a binary tree. St000516The number of stretching pairs of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001214The aft of an integer partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000650The number of 3-rises of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001552The number of inversions between excedances and fixed points of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001871The number of triconnected components of a graph. St001964The interval resolution global dimension of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000881The number of short braid edges in the graph of braid moves of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000570The Edelman-Greene number of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001220The width of a permutation. St001731The factorization defect of a permutation. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001715The number of non-records in a permutation. St001481The minimal height of a peak of a Dyck path. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001856The number of edges in the reduced word graph of a permutation. 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). St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000951The dimension of Ext^{1}(D(A),A) of the corresponding LNakayama algebra. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000335The difference of lower and upper interactions. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001389The number of partitions of the same length below the given integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001571The Cartan determinant of the integer partition. St001060The distinguishing index of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001520The number of strict 3-descents. St001586The number of odd parts smaller than the largest even part in an integer partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000068The number of minimal elements in a poset. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000379The number of Hamiltonian cycles in a graph. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000836The number of descents of distance 2 of a permutation. St000886The number of permutations with the same antidiagonal sums. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001924The number of cells in an integer partition whose arm and leg length coincide. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. 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). 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,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension 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. St001188The number of simple modules S with grade \inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \} at least two in the Nakayama algebra A corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St001432The order dimension of the partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000699The toughness times the least common multiple of 1,. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001071The beta invariant of the graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000451The length of the longest pattern of the form k 1 2. St000219The number of occurrences of the pattern 231 in a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000058The order of a permutation. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000264The girth of a graph, which is not a tree. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000338The number of pixed points of a permutation. St000462The major index minus the number of excedences of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000732The number of double deficiencies of a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001176The size of a partition minus its first part. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001961The sum of the greatest common divisors of all pairs of parts. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001933The largest multiplicity of a part in an integer partition. St000084The number of subtrees. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001498The normalised height of a Nakayama algebra with magnitude 1. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000941The number of characters of the symmetric group whose value on the partition is even. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001651The Frankl number of a lattice. St000735The last entry on the main diagonal of a standard tableau. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000102The charge of a semistandard tableau. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001556The number of inversions of the third entry of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001948The number of augmented double ascents of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001569The maximal modular displacement of a permutation. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type.
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