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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000292
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 1 = 0 + 1
011 => 1 = 0 + 1
0001 => 1 = 0 + 1
0010 => 1 = 0 + 1
0011 => 1 = 0 + 1
0101 => 2 = 1 + 1
0110 => 1 = 0 + 1
0111 => 1 = 0 + 1
1001 => 1 = 0 + 1
1011 => 1 = 0 + 1
00001 => 1 = 0 + 1
00010 => 1 = 0 + 1
00011 => 1 = 0 + 1
00100 => 1 = 0 + 1
00101 => 2 = 1 + 1
00110 => 1 = 0 + 1
00111 => 1 = 0 + 1
01001 => 2 = 1 + 1
01010 => 2 = 1 + 1
01011 => 2 = 1 + 1
01100 => 1 = 0 + 1
01101 => 2 = 1 + 1
01110 => 1 = 0 + 1
01111 => 1 = 0 + 1
10001 => 1 = 0 + 1
10010 => 1 = 0 + 1
10011 => 1 = 0 + 1
10101 => 2 = 1 + 1
10110 => 1 = 0 + 1
10111 => 1 = 0 + 1
11001 => 1 = 0 + 1
11011 => 1 = 0 + 1
000001 => 1 = 0 + 1
000010 => 1 = 0 + 1
000011 => 1 = 0 + 1
000100 => 1 = 0 + 1
000101 => 2 = 1 + 1
000110 => 1 = 0 + 1
000111 => 1 = 0 + 1
001000 => 1 = 0 + 1
001001 => 2 = 1 + 1
001010 => 2 = 1 + 1
001011 => 2 = 1 + 1
001100 => 1 = 0 + 1
001101 => 2 = 1 + 1
001110 => 1 = 0 + 1
001111 => 1 = 0 + 1
010001 => 2 = 1 + 1
010010 => 2 = 1 + 1
010011 => 2 = 1 + 1
Description
The number of ascents of a binary word.
Matching statistic: St000291
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 001 => 0
011 => 011 => 0
0001 => 0001 => 0
0010 => 0001 => 0
0011 => 0011 => 0
0101 => 0101 => 1
0110 => 0011 => 0
0111 => 0111 => 0
1001 => 0011 => 0
1011 => 0111 => 0
00001 => 00001 => 0
00010 => 00001 => 0
00011 => 00011 => 0
00100 => 00001 => 0
00101 => 00101 => 1
00110 => 00011 => 0
00111 => 00111 => 0
01001 => 00101 => 1
01010 => 00101 => 1
01011 => 01011 => 1
01100 => 00011 => 0
01101 => 01011 => 1
01110 => 00111 => 0
01111 => 01111 => 0
10001 => 00011 => 0
10010 => 00011 => 0
10011 => 00111 => 0
10101 => 01011 => 1
10110 => 00111 => 0
10111 => 01111 => 0
11001 => 00111 => 0
11011 => 01111 => 0
000001 => 000001 => 0
000010 => 000001 => 0
000011 => 000011 => 0
000100 => 000001 => 0
000101 => 000101 => 1
000110 => 000011 => 0
000111 => 000111 => 0
001000 => 000001 => 0
001001 => 001001 => 1
001010 => 000101 => 1
001011 => 001011 => 1
001100 => 000011 => 0
001101 => 001101 => 1
001110 => 000111 => 0
001111 => 001111 => 0
010001 => 000101 => 1
010010 => 000101 => 1
010011 => 001101 => 1
Description
The number of descents of a binary word.
Matching statistic: St000875
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 001 => 0
011 => 011 => 0
0001 => 0001 => 0
0010 => 0001 => 0
0011 => 0011 => 0
0101 => 0101 => 1
0110 => 0011 => 0
0111 => 0111 => 0
1001 => 0011 => 0
1011 => 0111 => 0
00001 => 00001 => 0
00010 => 00001 => 0
00011 => 00011 => 0
00100 => 00001 => 0
00101 => 00101 => 1
00110 => 00011 => 0
00111 => 00111 => 0
01001 => 00101 => 1
01010 => 00101 => 1
01011 => 01011 => 1
01100 => 00011 => 0
01101 => 01011 => 1
01110 => 00111 => 0
01111 => 01111 => 0
10001 => 00011 => 0
10010 => 00011 => 0
10011 => 00111 => 0
10101 => 01011 => 1
10110 => 00111 => 0
10111 => 01111 => 0
11001 => 00111 => 0
11011 => 01111 => 0
000001 => 000001 => 0
000010 => 000001 => 0
000011 => 000011 => 0
000100 => 000001 => 0
000101 => 000101 => 1
000110 => 000011 => 0
000111 => 000111 => 0
001000 => 000001 => 0
001001 => 001001 => 1
001010 => 000101 => 1
001011 => 001011 => 1
001100 => 000011 => 0
001101 => 001101 => 1
001110 => 000111 => 0
001111 => 001111 => 0
010001 => 000101 => 1
010010 => 000101 => 1
010011 => 001101 => 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as (D0)ℓD(1D)m, where D is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001421
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 001 => 0
011 => 011 => 0
0001 => 0001 => 0
0010 => 0001 => 0
0011 => 0011 => 0
0101 => 0101 => 1
0110 => 0011 => 0
0111 => 0111 => 0
1001 => 0011 => 0
1011 => 0111 => 0
00001 => 00001 => 0
00010 => 00001 => 0
00011 => 00011 => 0
00100 => 00001 => 0
00101 => 00101 => 1
00110 => 00011 => 0
00111 => 00111 => 0
01001 => 00101 => 1
01010 => 00101 => 1
01011 => 01011 => 1
01100 => 00011 => 0
01101 => 01011 => 1
01110 => 00111 => 0
01111 => 01111 => 0
10001 => 00011 => 0
10010 => 00011 => 0
10011 => 00111 => 0
10101 => 01011 => 1
10110 => 00111 => 0
10111 => 01111 => 0
11001 => 00111 => 0
11011 => 01111 => 0
000001 => 000001 => 0
000010 => 000001 => 0
000011 => 000011 => 0
000100 => 000001 => 0
000101 => 000101 => 1
000110 => 000011 => 0
000111 => 000111 => 0
001000 => 000001 => 0
001001 => 001001 => 1
001010 => 000101 => 1
001011 => 001011 => 1
001100 => 000011 => 0
001101 => 001101 => 1
001110 => 000111 => 0
001111 => 001111 => 0
010001 => 000101 => 1
010010 => 000101 => 1
010011 => 001101 => 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St000390
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 001 => 1 = 0 + 1
011 => 011 => 1 = 0 + 1
0001 => 0001 => 1 = 0 + 1
0010 => 0001 => 1 = 0 + 1
0011 => 0011 => 1 = 0 + 1
0101 => 0101 => 2 = 1 + 1
0110 => 0011 => 1 = 0 + 1
0111 => 0111 => 1 = 0 + 1
1001 => 0011 => 1 = 0 + 1
1011 => 0111 => 1 = 0 + 1
00001 => 00001 => 1 = 0 + 1
00010 => 00001 => 1 = 0 + 1
00011 => 00011 => 1 = 0 + 1
00100 => 00001 => 1 = 0 + 1
00101 => 00101 => 2 = 1 + 1
00110 => 00011 => 1 = 0 + 1
00111 => 00111 => 1 = 0 + 1
01001 => 00101 => 2 = 1 + 1
01010 => 00101 => 2 = 1 + 1
01011 => 01011 => 2 = 1 + 1
01100 => 00011 => 1 = 0 + 1
01101 => 01011 => 2 = 1 + 1
01110 => 00111 => 1 = 0 + 1
01111 => 01111 => 1 = 0 + 1
10001 => 00011 => 1 = 0 + 1
10010 => 00011 => 1 = 0 + 1
10011 => 00111 => 1 = 0 + 1
10101 => 01011 => 2 = 1 + 1
10110 => 00111 => 1 = 0 + 1
10111 => 01111 => 1 = 0 + 1
11001 => 00111 => 1 = 0 + 1
11011 => 01111 => 1 = 0 + 1
000001 => 000001 => 1 = 0 + 1
000010 => 000001 => 1 = 0 + 1
000011 => 000011 => 1 = 0 + 1
000100 => 000001 => 1 = 0 + 1
000101 => 000101 => 2 = 1 + 1
000110 => 000011 => 1 = 0 + 1
000111 => 000111 => 1 = 0 + 1
001000 => 000001 => 1 = 0 + 1
001001 => 001001 => 2 = 1 + 1
001010 => 000101 => 2 = 1 + 1
001011 => 001011 => 2 = 1 + 1
001100 => 000011 => 1 = 0 + 1
001101 => 001101 => 2 = 1 + 1
001110 => 000111 => 1 = 0 + 1
001111 => 001111 => 1 = 0 + 1
010001 => 000101 => 2 = 1 + 1
010010 => 000101 => 2 = 1 + 1
010011 => 001101 => 2 = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001267
Mp00224: Binary words —runsort⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00096: Binary words —Foata bijection⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 001 => 001 => 1 = 0 + 1
011 => 011 => 011 => 1 = 0 + 1
0001 => 0001 => 0001 => 1 = 0 + 1
0010 => 0001 => 0001 => 1 = 0 + 1
0011 => 0011 => 0011 => 1 = 0 + 1
0101 => 0101 => 1001 => 2 = 1 + 1
0110 => 0011 => 0011 => 1 = 0 + 1
0111 => 0111 => 0111 => 1 = 0 + 1
1001 => 0011 => 0011 => 1 = 0 + 1
1011 => 0111 => 0111 => 1 = 0 + 1
00001 => 00001 => 00001 => 1 = 0 + 1
00010 => 00001 => 00001 => 1 = 0 + 1
00011 => 00011 => 00011 => 1 = 0 + 1
00100 => 00001 => 00001 => 1 = 0 + 1
00101 => 00101 => 10001 => 2 = 1 + 1
00110 => 00011 => 00011 => 1 = 0 + 1
00111 => 00111 => 00111 => 1 = 0 + 1
01001 => 00101 => 10001 => 2 = 1 + 1
01010 => 00101 => 10001 => 2 = 1 + 1
01011 => 01011 => 10011 => 2 = 1 + 1
01100 => 00011 => 00011 => 1 = 0 + 1
01101 => 01011 => 10011 => 2 = 1 + 1
01110 => 00111 => 00111 => 1 = 0 + 1
01111 => 01111 => 01111 => 1 = 0 + 1
10001 => 00011 => 00011 => 1 = 0 + 1
10010 => 00011 => 00011 => 1 = 0 + 1
10011 => 00111 => 00111 => 1 = 0 + 1
10101 => 01011 => 10011 => 2 = 1 + 1
10110 => 00111 => 00111 => 1 = 0 + 1
10111 => 01111 => 01111 => 1 = 0 + 1
11001 => 00111 => 00111 => 1 = 0 + 1
11011 => 01111 => 01111 => 1 = 0 + 1
000001 => 000001 => 000001 => 1 = 0 + 1
000010 => 000001 => 000001 => 1 = 0 + 1
000011 => 000011 => 000011 => 1 = 0 + 1
000100 => 000001 => 000001 => 1 = 0 + 1
000101 => 000101 => 100001 => 2 = 1 + 1
000110 => 000011 => 000011 => 1 = 0 + 1
000111 => 000111 => 000111 => 1 = 0 + 1
001000 => 000001 => 000001 => 1 = 0 + 1
001001 => 001001 => 010001 => 2 = 1 + 1
001010 => 000101 => 100001 => 2 = 1 + 1
001011 => 001011 => 100011 => 2 = 1 + 1
001100 => 000011 => 000011 => 1 = 0 + 1
001101 => 001101 => 100101 => 2 = 1 + 1
001110 => 000111 => 000111 => 1 = 0 + 1
001111 => 001111 => 001111 => 1 = 0 + 1
010001 => 000101 => 100001 => 2 = 1 + 1
010010 => 000101 => 100001 => 2 = 1 + 1
010011 => 001101 => 100101 => 2 = 1 + 1
Description
The length of the Lyndon factorization of the binary word.
The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., w=l1…ln where each li is a Lyndon word and l1≥⋯≥ln.
Matching statistic: St001124
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 0
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 0
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 0
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 0
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 0
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
000001 => [6,1] => [[6,6],[5]]
=> [5]
=> 0
000010 => [5,2] => [[6,5],[4]]
=> [4]
=> 0
000011 => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
000100 => [4,3] => [[6,4],[3]]
=> [3]
=> 0
000101 => [4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> 1
000110 => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 0
001000 => [3,4] => [[6,3],[2]]
=> [2]
=> 0
001001 => [3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> 1
001010 => [3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> 1
001100 => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> 1
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 0
010001 => [2,4,1] => [[5,5,2],[4,1]]
=> [4,1]
=> 1
010010 => [2,3,2] => [[5,4,2],[3,1]]
=> [3,1]
=> 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity gλμ,ν of the Specht module Sλ in Sμ⊗Sν:
Sμ⊗Sν=⨁λgλμ,νSλ
This statistic records the Kronecker coefficient g(n−1)1λ,λ, for λ⊢n>1. For n≤1 the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000159
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1 = 0 + 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1 = 0 + 1
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 1 = 0 + 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 0 + 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1 = 0 + 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1 = 0 + 1
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 1 = 0 + 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1 = 0 + 1
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 1 = 0 + 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2 = 1 + 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1 = 0 + 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1 = 0 + 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2 = 1 + 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2 = 1 + 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2 = 1 + 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 0 + 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1 = 0 + 1
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1 = 0 + 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1 = 0 + 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 2 = 1 + 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1 = 0 + 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
000001 => [6,1] => [[6,6],[5]]
=> [5]
=> 1 = 0 + 1
000010 => [5,2] => [[6,5],[4]]
=> [4]
=> 1 = 0 + 1
000011 => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 1 = 0 + 1
000100 => [4,3] => [[6,4],[3]]
=> [3]
=> 1 = 0 + 1
000101 => [4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> 2 = 1 + 1
000110 => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 1 = 0 + 1
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 1 = 0 + 1
001000 => [3,4] => [[6,3],[2]]
=> [2]
=> 1 = 0 + 1
001001 => [3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> 2 = 1 + 1
001010 => [3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> 2 = 1 + 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> 2 = 1 + 1
001100 => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1 = 0 + 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> 2 = 1 + 1
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1 = 0 + 1
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 1 = 0 + 1
010001 => [2,4,1] => [[5,5,2],[4,1]]
=> [4,1]
=> 2 = 1 + 1
010010 => [2,3,2] => [[5,4,2],[3,1]]
=> [3,1]
=> 2 = 1 + 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> 2 = 1 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St001280
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => 001 => [3,1] => [3,1]
=> 1 = 0 + 1
011 => 011 => [2,1,1] => [2,1,1]
=> 1 = 0 + 1
0001 => 0001 => [4,1] => [4,1]
=> 1 = 0 + 1
0010 => 0001 => [4,1] => [4,1]
=> 1 = 0 + 1
0011 => 0011 => [3,1,1] => [3,1,1]
=> 1 = 0 + 1
0101 => 0101 => [2,2,1] => [2,2,1]
=> 2 = 1 + 1
0110 => 0011 => [3,1,1] => [3,1,1]
=> 1 = 0 + 1
0111 => 0111 => [2,1,1,1] => [2,1,1,1]
=> 1 = 0 + 1
1001 => 0011 => [3,1,1] => [3,1,1]
=> 1 = 0 + 1
1011 => 0111 => [2,1,1,1] => [2,1,1,1]
=> 1 = 0 + 1
00001 => 00001 => [5,1] => [5,1]
=> 1 = 0 + 1
00010 => 00001 => [5,1] => [5,1]
=> 1 = 0 + 1
00011 => 00011 => [4,1,1] => [4,1,1]
=> 1 = 0 + 1
00100 => 00001 => [5,1] => [5,1]
=> 1 = 0 + 1
00101 => 00101 => [3,2,1] => [3,2,1]
=> 2 = 1 + 1
00110 => 00011 => [4,1,1] => [4,1,1]
=> 1 = 0 + 1
00111 => 00111 => [3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
01001 => 00101 => [3,2,1] => [3,2,1]
=> 2 = 1 + 1
01010 => 00101 => [3,2,1] => [3,2,1]
=> 2 = 1 + 1
01011 => 01011 => [2,2,1,1] => [2,2,1,1]
=> 2 = 1 + 1
01100 => 00011 => [4,1,1] => [4,1,1]
=> 1 = 0 + 1
01101 => 01011 => [2,2,1,1] => [2,2,1,1]
=> 2 = 1 + 1
01110 => 00111 => [3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
01111 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1 = 0 + 1
10001 => 00011 => [4,1,1] => [4,1,1]
=> 1 = 0 + 1
10010 => 00011 => [4,1,1] => [4,1,1]
=> 1 = 0 + 1
10011 => 00111 => [3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
10101 => 01011 => [2,2,1,1] => [2,2,1,1]
=> 2 = 1 + 1
10110 => 00111 => [3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
10111 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1 = 0 + 1
11001 => 00111 => [3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
11011 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1 = 0 + 1
000001 => 000001 => [6,1] => [6,1]
=> 1 = 0 + 1
000010 => 000001 => [6,1] => [6,1]
=> 1 = 0 + 1
000011 => 000011 => [5,1,1] => [5,1,1]
=> 1 = 0 + 1
000100 => 000001 => [6,1] => [6,1]
=> 1 = 0 + 1
000101 => 000101 => [4,2,1] => [4,2,1]
=> 2 = 1 + 1
000110 => 000011 => [5,1,1] => [5,1,1]
=> 1 = 0 + 1
000111 => 000111 => [4,1,1,1] => [4,1,1,1]
=> 1 = 0 + 1
001000 => 000001 => [6,1] => [6,1]
=> 1 = 0 + 1
001001 => 001001 => [3,3,1] => [3,3,1]
=> 2 = 1 + 1
001010 => 000101 => [4,2,1] => [4,2,1]
=> 2 = 1 + 1
001011 => 001011 => [3,2,1,1] => [3,2,1,1]
=> 2 = 1 + 1
001100 => 000011 => [5,1,1] => [5,1,1]
=> 1 = 0 + 1
001101 => 001101 => [3,1,2,1] => [3,2,1,1]
=> 2 = 1 + 1
001110 => 000111 => [4,1,1,1] => [4,1,1,1]
=> 1 = 0 + 1
001111 => 001111 => [3,1,1,1,1] => [3,1,1,1,1]
=> 1 = 0 + 1
010001 => 000101 => [4,2,1] => [4,2,1]
=> 2 = 1 + 1
010010 => 000101 => [4,2,1] => [4,2,1]
=> 2 = 1 + 1
010011 => 001101 => [3,1,2,1] => [3,2,1,1]
=> 2 = 1 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000318
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2 = 0 + 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 2 = 0 + 2
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 2 = 0 + 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 0 + 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3 = 1 + 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2 = 0 + 2
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2 = 0 + 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2 = 0 + 2
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 2 = 0 + 2
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 2 = 0 + 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2 = 0 + 2
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 2 = 0 + 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 3 = 1 + 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2 = 0 + 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2 = 0 + 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3 = 1 + 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 3 = 1 + 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3 = 1 + 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3 = 1 + 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2 = 0 + 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 2 = 0 + 2
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2 = 0 + 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2 = 0 + 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2 = 0 + 2
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 3 = 1 + 2
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2 = 0 + 2
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2 = 0 + 2
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2 = 0 + 2
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2 = 0 + 2
000001 => [6,1] => [[6,6],[5]]
=> [5]
=> 2 = 0 + 2
000010 => [5,2] => [[6,5],[4]]
=> [4]
=> 2 = 0 + 2
000011 => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2 = 0 + 2
000100 => [4,3] => [[6,4],[3]]
=> [3]
=> 2 = 0 + 2
000101 => [4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> 3 = 1 + 2
000110 => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2 = 0 + 2
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 2 = 0 + 2
001000 => [3,4] => [[6,3],[2]]
=> [2]
=> 2 = 0 + 2
001001 => [3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> 3 = 1 + 2
001010 => [3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> 3 = 1 + 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> 3 = 1 + 2
001100 => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2 = 0 + 2
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> 3 = 1 + 2
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2 = 0 + 2
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 2 = 0 + 2
010001 => [2,4,1] => [[5,5,2],[4,1]]
=> [4,1]
=> 3 = 1 + 2
010010 => [2,3,2] => [[5,4,2],[3,1]]
=> [3,1]
=> 3 = 1 + 2
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> 3 = 1 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000767The number of runs in an integer composition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000386The number of factors DDU in a Dyck path. St000201The number of leaf nodes in a binary tree. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000068The number of minimal elements in a poset. St000568The hook number of a binary tree. St001354The number of series nodes in the modular decomposition of a graph. St000356The number of occurrences of the pattern 13-2. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000092The number of outer peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000834The number of right outer peaks of a permutation.
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