Processing math: 100%

Your data matches 41 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000664
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 0
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 0
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 0
=> [1] => [1,0]
 => [2,1] => 0
Description
The number of right ropes of a permutation. Let π be a permutation of length n. A raft of π is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of π. See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000761
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 0
1 => [1] => [1] => 0
00 => [2] => [1,1] => 0
01 => [1,1] => [2] => 0
10 => [1,1] => [2] => 0
11 => [2] => [1,1] => 0
000 => [3] => [1,1,1] => 0
001 => [2,1] => [1,2] => 1
010 => [1,1,1] => [3] => 0
011 => [1,2] => [2,1] => 0
100 => [1,2] => [2,1] => 0
101 => [1,1,1] => [3] => 0
110 => [2,1] => [1,2] => 1
111 => [3] => [1,1,1] => 0
0000 => [4] => [1,1,1,1] => 0
0001 => [3,1] => [1,1,2] => 1
0010 => [2,1,1] => [1,3] => 1
0011 => [2,2] => [1,2,1] => 1
0100 => [1,1,2] => [3,1] => 0
0101 => [1,1,1,1] => [4] => 0
0110 => [1,2,1] => [2,2] => 0
0111 => [1,3] => [2,1,1] => 0
1000 => [1,3] => [2,1,1] => 0
1001 => [1,2,1] => [2,2] => 0
1010 => [1,1,1,1] => [4] => 0
1011 => [1,1,2] => [3,1] => 0
1100 => [2,2] => [1,2,1] => 1
1101 => [2,1,1] => [1,3] => 1
1110 => [3,1] => [1,1,2] => 1
1111 => [4] => [1,1,1,1] => 0
=> [] => [] => ? = 0
Description
The number of ascents in an integer composition. A composition has an ascent, or rise, at position i if ai<ai+1.
Matching statistic: St000760
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 1 = 0 + 1
1 => [1] => [1] => 1 = 0 + 1
00 => [2] => [1,1] => 1 = 0 + 1
01 => [1,1] => [2] => 1 = 0 + 1
10 => [1,1] => [2] => 1 = 0 + 1
11 => [2] => [1,1] => 1 = 0 + 1
000 => [3] => [1,1,1] => 1 = 0 + 1
001 => [2,1] => [2,1] => 2 = 1 + 1
010 => [1,1,1] => [3] => 1 = 0 + 1
011 => [1,2] => [1,2] => 1 = 0 + 1
100 => [1,2] => [1,2] => 1 = 0 + 1
101 => [1,1,1] => [3] => 1 = 0 + 1
110 => [2,1] => [2,1] => 2 = 1 + 1
111 => [3] => [1,1,1] => 1 = 0 + 1
0000 => [4] => [1,1,1,1] => 1 = 0 + 1
0001 => [3,1] => [2,1,1] => 2 = 1 + 1
0010 => [2,1,1] => [3,1] => 2 = 1 + 1
0011 => [2,2] => [1,2,1] => 2 = 1 + 1
0100 => [1,1,2] => [1,3] => 1 = 0 + 1
0101 => [1,1,1,1] => [4] => 1 = 0 + 1
0110 => [1,2,1] => [2,2] => 1 = 0 + 1
0111 => [1,3] => [1,1,2] => 1 = 0 + 1
1000 => [1,3] => [1,1,2] => 1 = 0 + 1
1001 => [1,2,1] => [2,2] => 1 = 0 + 1
1010 => [1,1,1,1] => [4] => 1 = 0 + 1
1011 => [1,1,2] => [1,3] => 1 = 0 + 1
1100 => [2,2] => [1,2,1] => 2 = 1 + 1
1101 => [2,1,1] => [3,1] => 2 = 1 + 1
1110 => [3,1] => [2,1,1] => 2 = 1 + 1
1111 => [4] => [1,1,1,1] => 1 = 0 + 1
=> [] => [] => ? = 0 + 1
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Matching statistic: St000764
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 1 = 0 + 1
1 => [1] => [1] => 1 = 0 + 1
00 => [2] => [1,1] => 1 = 0 + 1
01 => [1,1] => [2] => 1 = 0 + 1
10 => [1,1] => [2] => 1 = 0 + 1
11 => [2] => [1,1] => 1 = 0 + 1
000 => [3] => [1,1,1] => 1 = 0 + 1
001 => [2,1] => [1,2] => 2 = 1 + 1
010 => [1,1,1] => [3] => 1 = 0 + 1
011 => [1,2] => [2,1] => 1 = 0 + 1
100 => [1,2] => [2,1] => 1 = 0 + 1
101 => [1,1,1] => [3] => 1 = 0 + 1
110 => [2,1] => [1,2] => 2 = 1 + 1
111 => [3] => [1,1,1] => 1 = 0 + 1
0000 => [4] => [1,1,1,1] => 1 = 0 + 1
0001 => [3,1] => [1,1,2] => 2 = 1 + 1
0010 => [2,1,1] => [1,3] => 2 = 1 + 1
0011 => [2,2] => [1,2,1] => 2 = 1 + 1
0100 => [1,1,2] => [3,1] => 1 = 0 + 1
0101 => [1,1,1,1] => [4] => 1 = 0 + 1
0110 => [1,2,1] => [2,2] => 1 = 0 + 1
0111 => [1,3] => [2,1,1] => 1 = 0 + 1
1000 => [1,3] => [2,1,1] => 1 = 0 + 1
1001 => [1,2,1] => [2,2] => 1 = 0 + 1
1010 => [1,1,1,1] => [4] => 1 = 0 + 1
1011 => [1,1,2] => [3,1] => 1 = 0 + 1
1100 => [2,2] => [1,2,1] => 2 = 1 + 1
1101 => [2,1,1] => [1,3] => 2 = 1 + 1
1110 => [3,1] => [1,1,2] => 2 = 1 + 1
1111 => [4] => [1,1,1,1] => 1 = 0 + 1
=> [] => [] => ? = 0 + 1
Description
The number of strong records in an integer composition. A strong record is an element ai such that ai>aj for all j<i. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St000552
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
 => 0
1 => [1] => [1] => ([],1)
 => 0
00 => [2] => [2] => ([],2)
 => 0
01 => [1,1] => [1,1] => ([(0,1)],2)
 => 0
10 => [1,1] => [1,1] => ([(0,1)],2)
 => 0
11 => [2] => [2] => ([],2)
 => 0
000 => [3] => [3] => ([],3)
 => 0
001 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
010 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
011 => [1,2] => [1,2] => ([(1,2)],3)
 => 0
100 => [1,2] => [1,2] => ([(1,2)],3)
 => 0
101 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
110 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
111 => [3] => [3] => ([],3)
 => 0
0000 => [4] => [4] => ([],4)
 => 0
0001 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
0010 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
0011 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 1
0100 => [1,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
0101 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
0110 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
0111 => [1,3] => [1,3] => ([(2,3)],4)
 => 0
1000 => [1,3] => [1,3] => ([(2,3)],4)
 => 0
1001 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1010 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1011 => [1,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
1100 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 1
1101 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
1110 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
1111 => [4] => [4] => ([],4)
 => 0
=> [] => ? => ?
 => ? = 0
Description
The number of cut vertices of a graph. A cut vertex is one whose deletion increases the number of connected components.
Matching statistic: St000660
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => [1,0]
 => 0
1 => [1] => [1,0]
 => [1,0]
 => 0
00 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
01 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
10 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
11 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
000 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
001 => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 0
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 0
110 => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1
111 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
=> [] => ?
 => ?
 => ? = 0
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001085
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => [1] => 0
1 => [1] => [1,0]
 => [1] => 0
00 => [2] => [1,1,0,0]
 => [2,1] => 0
01 => [1,1] => [1,0,1,0]
 => [1,2] => 0
10 => [1,1] => [1,0,1,0]
 => [1,2] => 0
11 => [2] => [1,1,0,0]
 => [2,1] => 0
000 => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
001 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
011 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 0
100 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
110 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
111 => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
=> [] => ?
 => ? => ? = 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern 213, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St000701
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => [.,.]
 => 1 = 0 + 1
1 => [1] => [1,0]
 => [.,.]
 => 1 = 0 + 1
00 => [2] => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
01 => [1,1] => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
10 => [1,1] => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
11 => [2] => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
000 => [3] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
011 => [1,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
100 => [1,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
110 => [2,1] => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 1 = 0 + 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 1 = 0 + 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
=> [] => ?
 => ?
 => ? = 0 + 1
Description
The protection number of a binary tree. This is the minimal distance from the root to a leaf.
Matching statistic: St001732
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
1 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
00 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
01 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
10 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
11 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
000 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
110 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
=> [] => ?
 => ?
 => ? = 0 + 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St001677
(load all 4 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001677: Lattices ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
0 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
1 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
00 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 0
11 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
000 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 0
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 0
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
 => 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1
111 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 0
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ([(0,8),(2,11),(3,11),(4,9),(5,9),(6,10),(7,10),(8,2),(8,3),(9,1),(10,4),(10,5),(11,6),(11,7)],12)
 => ? = 0
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ([(0,9),(2,17),(3,11),(3,15),(4,11),(4,14),(5,12),(6,13),(7,10),(8,7),(8,17),(9,2),(9,8),(10,14),(10,15),(11,18),(12,16),(13,16),(14,5),(14,18),(15,6),(15,18),(16,1),(17,3),(17,4),(17,10),(18,12),(18,13)],19)
 => ? = 0
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 0
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 0
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ([(0,9),(2,17),(3,11),(3,15),(4,11),(4,14),(5,12),(6,13),(7,10),(8,7),(8,17),(9,2),(9,8),(10,14),(10,15),(11,18),(12,16),(13,16),(14,5),(14,18),(15,6),(15,18),(16,1),(17,3),(17,4),(17,10),(18,12),(18,13)],19)
 => ? = 0
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ([(0,8),(2,11),(3,11),(4,9),(5,9),(6,10),(7,10),(8,2),(8,3),(9,1),(10,4),(10,5),(11,6),(11,7)],12)
 => ? = 0
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 0
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
 => ? = 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(0,9),(2,14),(2,15),(3,13),(3,14),(4,17),(5,12),(6,11),(7,10),(8,7),(8,17),(9,4),(9,8),(10,13),(10,15),(11,16),(12,16),(13,18),(14,6),(14,18),(15,5),(15,18),(16,1),(17,2),(17,3),(17,10),(18,11),(18,12)],19)
 => ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
=> ?
 => ?
 => ? = 0
Description
The number of non-degenerate subsets of a lattice whose meet is the bottom element. A subset whose meet is the bottom element is non-degenerate, if it neither contains the bottom, nor the top element of the lattice.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001722The number of minimal chains with small intervals between a binary word and the top element. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.