searching the database
Your data matches 434 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001240
(load all 51 compositions to match this statistic)
(load all 51 compositions to match this statistic)
St001240: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
Matching statistic: St000967
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
St000967: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> 3 = 1 + 2
[1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0]
=> 6 = 4 + 2
[1,0,1,0,1,1,0,0]
=> 6 = 4 + 2
[1,0,1,1,0,0,1,0]
=> 6 = 4 + 2
[1,0,1,1,1,0,0,0]
=> 6 = 4 + 2
[1,1,0,0,1,0,1,0]
=> 6 = 4 + 2
[1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[1,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0]
=> 6 = 4 + 2
[1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0]
=> 7 = 5 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 6 + 2
Description
The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra.
Matching statistic: St000144
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000144: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000144: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000998
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000998: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000998: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001012
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St001012: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001012: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000031
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00223: Permutations ārunsortā¶ Permutations
St000031: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00223: Permutations ārunsortā¶ Permutations
St000031: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000056
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00223: Permutations ārunsortā¶ Permutations
St000056: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00223: Permutations ārunsortā¶ Permutations
St000056: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
Matching statistic: St000213
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00223: Permutations ārunsortā¶ Permutations
St000213: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00223: Permutations ārunsortā¶ Permutations
St000213: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000314
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00223: Permutations ārunsortā¶ Permutations
St000314: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00223: Permutations ārunsortā¶ Permutations
St000314: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000636
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1,0]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,1,0,0]
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 3
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,1,1,1,0,0,0,0]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
Description
The hull number of a graph.
The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$.
The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.
The following 424 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000907The number of maximal antichains of minimal length in a poset. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000234The number of global ascents of a permutation. St001405The number of bonds in a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000053The number of valleys of the Dyck path. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000203The number of external nodes of a binary tree. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000308The height of the tree associated to a permutation. St000315The number of isolated vertices of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000740The last entry of a permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000839The largest opener of a set partition. St000876The number of factors in the Catalan decomposition of a binary word. St000916The packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St000992The alternating sum of the parts of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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:
St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001339The irredundance number of a graph. 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. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001437The flex of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000778The metric dimension of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001949The rigidity index of a graph. 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. St000064The number of one-box pattern of a permutation. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000060The greater neighbor of the maximum. St000627The exponent of a binary word. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000051The size of the left subtree of a binary tree. St000134The size of the orbit of an alternating sign matrix under gyration. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000210Minimum over maximum difference of elements in cycles. St000326The position of the first one in a binary word after appending a 1 at the end. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000485The length of the longest cycle of a permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000831The number of indices that are either descents or recoils. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000906The length of the shortest maximal chain in a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001115The number of even descents of a permutation. 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. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001480The number of simple summands of the module J^2/J^3. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000054The first entry of the permutation. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000356The number of occurrences of the pattern 13-2. St000392The length of the longest run of ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000530The number of permutations with the same descent word as the given permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000543The size of the conjugacy class of a binary word. St000657The smallest part of an integer composition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000877The depth of the binary word interpreted as a path. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000982The length of the longest constant subword. St000989The number of final rises of a permutation. St001052The length of the exterior of a permutation. St001246The maximal difference between two consecutive entries of a permutation. 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. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000836The number of descents of distance 2 of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001552The number of inversions between excedances and fixed points of a permutation. St001948The number of augmented double ascents of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000480The number of lower covers of a partition in dominance order. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000327The number of cover relations in a poset. St000395The sum of the heights of the peaks of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules 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. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001875The number of simple modules with projective dimension at most 1. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000522The number of 1-protected nodes of a rooted tree. St001645The pebbling number of a connected graph. St000521The number of distinct subtrees of an ordered tree. St000050The depth or height of a binary tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001118The acyclic chromatic index of a graph. St001114The number of odd descents of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001928The number of non-overlapping descents in a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000216The absolute length of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000809The reduced reflection length of the permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000094The depth of an ordered tree. St000259The diameter of a connected graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001469The holeyness of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001822The number of alignments of a signed permutation. St001935The number of ascents in a parking function. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000488The number of cycles of a permutation of length at most 2. St000632The jump number of the poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001820The size of the image of the pop stack sorting operator. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000527The width of the poset. St000528The height of a poset. St000845The maximal number of elements covered by an element in a poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000643The size of the largest orbit of antichains under Panyushev complementation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001812The biclique partition number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001060The distinguishing index of a graph. St001555The order of a signed permutation. St001782The order of rowmotion on the set of order ideals of a poset. St001432The order dimension of the partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St000474Dyson's crank of a partition. 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 $eA$. St000454The largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001569The maximal modular displacement of a permutation. St000101The cocharge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001856The number of edges in the reduced word graph of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000116The major index of a semistandard tableau obtained by standardizing. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. 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. St000624The normalized sum of the minimal distances to a greater element. St000736The last entry in the first row of a semistandard tableau. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001596The number of two-by-two squares inside a skew partition. St001684The reduced word complexity 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. St001768The number of reduced words of a signed permutation. St001864The number of excedances of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000021The number of descents of a permutation. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000443The number of long tunnels of a Dyck path. St000461The rix statistic of a permutation. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000662The staircase size of the code of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001117The game chromatic index of a graph. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001964The interval resolution global dimension of a poset. St000135The number of lucky cars of the parking function. St000325The width of the tree associated to a permutation. St000456The monochromatic index of a connected graph. St000470The number of runs in a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000988The orbit size of a permutation under Foata's bijection. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000422The energy of a graph, if it is integral. St000735The last entry on the main diagonal of a standard tableau.
Sorry, this statistic was not found in the database
or
add this statistic to the database ā it's very simple and we need your support!