Your data matches 52 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000165
(load all 10 compositions to match this statistic)
Mapping
St000165: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[1,2] => 3
[2,1] => 3
[1,1,1] => 3
[1,1,2] => 4
[1,2,1] => 4
[2,1,1] => 4
[1,1,3] => 5
[1,3,1] => 5
[3,1,1] => 5
[1,2,2] => 5
[2,1,2] => 5
[2,2,1] => 5
[1,2,3] => 6
[1,3,2] => 6
[2,1,3] => 6
[2,3,1] => 6
[3,1,2] => 6
[3,2,1] => 6
Description
The sum of the entries of a parking function. The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
Matching statistic: St000103
(load all 4 compositions to match this statistic)
Mapping
Mp00302: Parking functions —insertion tableau⟶ Semistandard tableaux
St000103: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
 => 1
[1,1] => [[1,1]]
 => 2
[1,2] => [[1,2]]
 => 3
[2,1] => [[1],[2]]
 => 3
[1,1,1] => [[1,1,1]]
 => 3
[1,1,2] => [[1,1,2]]
 => 4
[1,2,1] => [[1,1],[2]]
 => 4
[2,1,1] => [[1,1],[2]]
 => 4
[1,1,3] => [[1,1,3]]
 => 5
[1,3,1] => [[1,1],[3]]
 => 5
[3,1,1] => [[1,1],[3]]
 => 5
[1,2,2] => [[1,2,2]]
 => 5
[2,1,2] => [[1,2],[2]]
 => 5
[2,2,1] => [[1,2],[2]]
 => 5
[1,2,3] => [[1,2,3]]
 => 6
[1,3,2] => [[1,2],[3]]
 => 6
[2,1,3] => [[1,3],[2]]
 => 6
[2,3,1] => [[1,3],[2]]
 => 6
[3,1,2] => [[1,2],[3]]
 => 6
[3,2,1] => [[1],[2],[3]]
 => 6
Description
The sum of the entries of a semistandard tableau.
Matching statistic: St000144
(load all 2 compositions to match this statistic)
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,1] => [1,1] => [1,0,1,0]
 => 2
[1,2] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,1] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 6
[1,3,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
[2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 6
[2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
[3,1,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 6
[3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 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: St000228
(load all 4 compositions to match this statistic)
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 1
[1,1] => [1,1] => [1,1]
 => 2
[1,2] => [1,2] => [2,1]
 => 3
[2,1] => [2,1] => [2,1]
 => 3
[1,1,1] => [1,1,1] => [1,1,1]
 => 3
[1,1,2] => [1,1,2] => [2,1,1]
 => 4
[1,2,1] => [1,2,1] => [2,1,1]
 => 4
[2,1,1] => [2,1,1] => [2,1,1]
 => 4
[1,1,3] => [1,1,3] => [3,1,1]
 => 5
[1,3,1] => [1,3,1] => [3,1,1]
 => 5
[3,1,1] => [3,1,1] => [3,1,1]
 => 5
[1,2,2] => [1,2,2] => [2,2,1]
 => 5
[2,1,2] => [2,1,2] => [2,2,1]
 => 5
[2,2,1] => [2,2,1] => [2,2,1]
 => 5
[1,2,3] => [1,2,3] => [3,2,1]
 => 6
[1,3,2] => [1,3,2] => [3,2,1]
 => 6
[2,1,3] => [2,1,3] => [3,2,1]
 => 6
[2,3,1] => [2,3,1] => [3,2,1]
 => 6
[3,1,2] => [3,1,2] => [3,2,1]
 => 6
[3,2,1] => [3,2,1] => [3,2,1]
 => 6
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000231
(load all 2 compositions to match this statistic)
Mapping
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
Mp00285: Ordered set partitions —to set partition⟶ Set partitions
St000231: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [{1}] => {{1}}
 => 1
[1,1] => [{1,2}] => {{1,2}}
 => 2
[1,2] => [{1},{2}] => {{1},{2}}
 => 3
[2,1] => [{2},{1}] => {{1},{2}}
 => 3
[1,1,1] => [{1,2,3}] => {{1,2,3}}
 => 3
[1,1,2] => [{1,2},{3}] => {{1,2},{3}}
 => 5
[1,2,1] => [{1,3},{2}] => {{1,3},{2}}
 => 5
[2,1,1] => [{2,3},{1}] => {{1},{2,3}}
 => 4
[1,1,3] => [{1,2},{3}] => {{1,2},{3}}
 => 5
[1,3,1] => [{1,3},{2}] => {{1,3},{2}}
 => 5
[3,1,1] => [{2,3},{1}] => {{1},{2,3}}
 => 4
[1,2,2] => [{1},{2,3}] => {{1},{2,3}}
 => 4
[2,1,2] => [{2},{1,3}] => {{1,3},{2}}
 => 5
[2,2,1] => [{3},{1,2}] => {{1,2},{3}}
 => 5
[1,2,3] => [{1},{2},{3}] => {{1},{2},{3}}
 => 6
[1,3,2] => [{1},{3},{2}] => {{1},{2},{3}}
 => 6
[2,1,3] => [{2},{1},{3}] => {{1},{2},{3}}
 => 6
[2,3,1] => [{3},{1},{2}] => {{1},{2},{3}}
 => 6
[3,1,2] => [{2},{3},{1}] => {{1},{2},{3}}
 => 6
[3,2,1] => [{3},{2},{1}] => {{1},{2},{3}}
 => 6
Description
Sum of the maximal elements of the blocks of a set partition.
Matching statistic: St000395
(load all 2 compositions to match this statistic)
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,1] => [1,1] => [1,0,1,0]
 => 2
[1,2] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,1] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 6
[1,3,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
[2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 6
[2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
[3,1,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 6
[3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000548
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 1
[1,1] => [1,1] => [1,1]
 => 2
[1,2] => [1,2] => [2,1]
 => 3
[2,1] => [2,1] => [2,1]
 => 3
[1,1,1] => [1,1,1] => [1,1,1]
 => 3
[1,1,2] => [1,1,2] => [2,1,1]
 => 4
[1,2,1] => [1,2,1] => [2,1,1]
 => 4
[2,1,1] => [2,1,1] => [2,1,1]
 => 4
[1,1,3] => [1,1,3] => [3,1,1]
 => 5
[1,3,1] => [1,3,1] => [3,1,1]
 => 5
[3,1,1] => [3,1,1] => [3,1,1]
 => 5
[1,2,2] => [1,2,2] => [2,2,1]
 => 5
[2,1,2] => [2,1,2] => [2,2,1]
 => 5
[2,2,1] => [2,2,1] => [2,2,1]
 => 5
[1,2,3] => [1,2,3] => [3,2,1]
 => 6
[1,3,2] => [1,3,2] => [3,2,1]
 => 6
[2,1,3] => [2,1,3] => [3,2,1]
 => 6
[2,3,1] => [2,3,1] => [3,2,1]
 => 6
[3,1,2] => [3,1,2] => [3,2,1]
 => 6
[3,2,1] => [3,2,1] => [3,2,1]
 => 6
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St001018
(load all 2 compositions to match this statistic)
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,1] => [1,1] => [1,0,1,0]
 => 2
[1,2] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,1] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 6
[1,3,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
[2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 6
[2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
[3,1,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 6
[3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001020
(load all 2 compositions to match this statistic)
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,1] => [1,1] => [1,0,1,0]
 => 2
[1,2] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,1] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5
[1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 6
[1,3,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6
[2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 6
[2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
[3,1,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 6
[3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000998
(load all 2 compositions to match this statistic)
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 2 = 1 + 1
[1,1] => [1,1] => [1,0,1,0]
 => 3 = 2 + 1
[1,2] => [1,2] => [1,0,1,1,0,0]
 => 4 = 3 + 1
[2,1] => [2,1] => [1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[2,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 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.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000018The number of inversions of a permutation. St000114The sum of the entries of the Gelfand-Tsetlin pattern. St000189The number of elements in the poset. St000224The sorting index of a permutation. St000293The number of inversions of a binary word. St000400The path length of an ordered tree. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000494The number of inversions of distance at most 3 of a permutation. St000738The first entry in the last row of a standard tableau. St000833The comajor index of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001034The area of the parallelogram polyomino associated with the Dyck path. St001161The major index north count of a Dyck path. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001342The number of vertices in the center of a graph. St001437The flex of a binary word. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St000639The number of relations in a poset. St000806The semiperimeter of the associated bargraph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000327The number of cover relations in a poset. St000422The energy of a graph, if it is integral. St001545The second Elser number of a connected graph. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001645The pebbling number of a connected graph. St000077The number of boxed and circled entries. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001407The number of minimal entries in a semistandard tableau.