Your data matches 320 different statistics following compositions of up to 3 maps.
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Matching statistic: St000628
(load all 122 compositions to match this statistic)
Mapping
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 1 = 2 - 1
111 => 0 = 1 - 1
0000 => 0 = 1 - 1
0001 => 1 = 2 - 1
0101 => 1 = 2 - 1
0111 => 1 = 2 - 1
1000 => 1 = 2 - 1
1010 => 1 = 2 - 1
1110 => 1 = 2 - 1
1111 => 0 = 1 - 1
00000 => 0 = 1 - 1
11111 => 0 = 1 - 1
000000 => 0 = 1 - 1
111111 => 0 = 1 - 1
0000000 => 0 = 1 - 1
1111111 => 0 = 1 - 1
00000000 => 0 = 1 - 1
11111111 => 0 = 1 - 1
000000000 => 0 = 1 - 1
111111111 => 0 = 1 - 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000630
(load all 19 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1
1 => 1 => 1
00 => 00 => 1
01 => 01 => 2
10 => 01 => 2
11 => 11 => 1
000 => 000 => 1
001 => 001 => 2
010 => 001 => 2
011 => 011 => 2
100 => 001 => 2
101 => 011 => 2
110 => 011 => 2
111 => 111 => 1
0000 => 0000 => 1
0001 => 0001 => 2
0101 => 0101 => 2
0111 => 0111 => 2
1000 => 0001 => 2
1010 => 0011 => 2
1110 => 0111 => 2
1111 => 1111 => 1
00000 => 00000 => 1
11111 => 11111 => 1
000000 => 000000 => 1
111111 => 111111 => 1
0000000 => 0000000 => 1
1111111 => 1111111 => 1
00000000 => 00000000 => 1
11111111 => 11111111 => 1
000000000 => 000000000 => 1
111111111 => 111111111 => 1
Description
The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St000983
(load all 13 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1
1 => 1 => 1
00 => 00 => 1
01 => 10 => 2
10 => 10 => 2
11 => 11 => 1
000 => 000 => 1
001 => 100 => 2
010 => 100 => 2
011 => 110 => 2
100 => 100 => 2
101 => 110 => 2
110 => 110 => 2
111 => 111 => 1
0000 => 0000 => 1
0001 => 1000 => 2
0101 => 1100 => 2
0111 => 1110 => 2
1000 => 1000 => 2
1010 => 1100 => 2
1110 => 1110 => 2
1111 => 1111 => 1
00000 => 00000 => 1
11111 => 11111 => 1
000000 => 000000 => 1
111111 => 111111 => 1
0000000 => 0000000 => 1
1111111 => 1111111 => 1
00000000 => 00000000 => 1
11111111 => 11111111 => 1
000000000 => 000000000 => 1
111111111 => 111111111 => 1
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000291
(load all 21 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 0 = 1 - 1
00 => 00 => 0 = 1 - 1
01 => 10 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 0 = 1 - 1
000 => 000 => 0 = 1 - 1
001 => 100 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 110 => 1 = 2 - 1
100 => 100 => 1 = 2 - 1
101 => 110 => 1 = 2 - 1
110 => 110 => 1 = 2 - 1
111 => 111 => 0 = 1 - 1
0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 1 = 2 - 1
0101 => 1100 => 1 = 2 - 1
0111 => 1110 => 1 = 2 - 1
1000 => 1000 => 1 = 2 - 1
1010 => 1100 => 1 = 2 - 1
1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 0 = 1 - 1
00000 => 00000 => 0 = 1 - 1
11111 => 11111 => 0 = 1 - 1
000000 => 000000 => 0 = 1 - 1
111111 => 111111 => 0 = 1 - 1
0000000 => 0000000 => 0 = 1 - 1
1111111 => 1111111 => 0 = 1 - 1
00000000 => 00000000 => 0 = 1 - 1
11111111 => 11111111 => 0 = 1 - 1
000000000 => 000000000 => 0 = 1 - 1
111111111 => 111111111 => 0 = 1 - 1
Description
The number of descents of a binary word.
Matching statistic: St000691
(load all 6 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 0 = 1 - 1
00 => 00 => 0 = 1 - 1
01 => 10 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 0 = 1 - 1
000 => 000 => 0 = 1 - 1
001 => 100 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 110 => 1 = 2 - 1
100 => 100 => 1 = 2 - 1
101 => 110 => 1 = 2 - 1
110 => 110 => 1 = 2 - 1
111 => 111 => 0 = 1 - 1
0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 1 = 2 - 1
0101 => 1100 => 1 = 2 - 1
0111 => 1110 => 1 = 2 - 1
1000 => 1000 => 1 = 2 - 1
1010 => 1100 => 1 = 2 - 1
1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 0 = 1 - 1
00000 => 00000 => 0 = 1 - 1
11111 => 11111 => 0 = 1 - 1
000000 => 000000 => 0 = 1 - 1
111111 => 111111 => 0 = 1 - 1
0000000 => 0000000 => 0 = 1 - 1
1111111 => 1111111 => 0 = 1 - 1
00000000 => 00000000 => 0 = 1 - 1
11111111 => 11111111 => 0 = 1 - 1
000000000 => 000000000 => 0 = 1 - 1
111111111 => 111111111 => 0 = 1 - 1
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001420
(load all 35 compositions to match this statistic)
Mapping
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001420: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 0 = 1 - 1
00 => 00 => 0 = 1 - 1
01 => 01 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 0 = 1 - 1
000 => 000 => 0 = 1 - 1
001 => 001 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 011 => 1 = 2 - 1
100 => 010 => 1 = 2 - 1
101 => 101 => 1 = 2 - 1
110 => 110 => 1 = 2 - 1
111 => 111 => 0 = 1 - 1
0000 => 0000 => 0 = 1 - 1
0001 => 0001 => 1 = 2 - 1
0101 => 1001 => 1 = 2 - 1
0111 => 0111 => 1 = 2 - 1
1000 => 0010 => 1 = 2 - 1
1010 => 0110 => 1 = 2 - 1
1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 0 = 1 - 1
00000 => 00000 => 0 = 1 - 1
11111 => 11111 => 0 = 1 - 1
000000 => 000000 => 0 = 1 - 1
111111 => 111111 => 0 = 1 - 1
0000000 => 0000000 => 0 = 1 - 1
1111111 => 1111111 => 0 = 1 - 1
00000000 => 00000000 => 0 = 1 - 1
11111111 => 11111111 => 0 = 1 - 1
000000000 => 000000000 => 0 = 1 - 1
111111111 => 111111111 => 0 = 1 - 1
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Matching statistic: St000010
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => [2]
 => 1
1 => ([(0,1)],2)
 => [2]
 => 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
11 => ([(0,2),(2,1)],3)
 => [3]
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
00000000 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
000000000 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
Description
The length of the partition.
Matching statistic: St000159
(load all 4 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => [2]
 => 1
1 => ([(0,1)],2)
 => [2]
 => 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
11 => ([(0,2),(2,1)],3)
 => [3]
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
00000000 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
000000000 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 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: St000278
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000278: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => [2]
 => 1
1 => ([(0,1)],2)
 => [2]
 => 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
11 => ([(0,2),(2,1)],3)
 => [3]
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
00000000 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
000000000 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$, where $k$ is the number of parts of $\lambda$. An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
Matching statistic: St000346
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => [2]
 => 1
1 => ([(0,1)],2)
 => [2]
 => 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
11 => ([(0,2),(2,1)],3)
 => [3]
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
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)
 => [5,3]
 => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 1
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 1
00000000 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 1
000000000 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 1
Description
The number of coarsenings of a partition. A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
The following 310 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000982The length of the longest constant subword. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001486The number of corners of the ribbon associated with an integer composition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000292The number of ascents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000481The number of upper covers of a partition in dominance order. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001673The degree of asymmetry of an integer composition. St000013The height of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000808The number of up steps of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St001128The exponens consonantiae of a partition. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001746The coalition number of a graph. St001814The number of partitions interlacing the given partition. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000183The side length of the Durfee square of an integer partition. St000225Difference between largest and smallest parts in a partition. St000290The major index of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000480The number of lower covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000662The staircase size of the code of a permutation. St000682The Grundy value of Welter's game on a binary word. St000703The number of deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000769The major index of a composition regarded as a word. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001092The number of distinct even parts of a partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001485The modular major index of a binary word. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000527The width of the poset. St000758The length of the longest staircase fitting into an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000761The number of ascents in an integer composition. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000340The number of non-final maximal constant sub-paths of length greater than one. St000097The order of the largest clique of the graph. St001261The Castelnuovo-Mumford regularity of a graph. St000535The rank-width of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000397The Strahler number of a rooted tree. St000444The length of the maximal rise of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001674The number of vertices of the largest induced star graph in the graph. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000919The number of maximal left branches of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001928The number of non-overlapping descents in a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000298The order dimension or Dushnik-Miller dimension of a poset. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001645The pebbling number of a connected graph. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000238The number of indices that are not small weak excedances. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000316The number of non-left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001271The competition number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001874Lusztig's a-function for the symmetric group. St001741The largest integer such that all patterns of this size are contained in the permutation. St001469The holeyness of a permutation. St000098The chromatic number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001734The lettericity of a graph. St000092The number of outer peaks of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000331The number of upper interactions of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001569The maximal modular displacement of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000307The number of rowmotion orbits of a poset. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001792The arboricity of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St001270The bandwidth of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001057The Grundy value of the game of creating an independent set in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000640The rank of the largest boolean interval in a poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001642The Prague dimension of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001323The independence gap of a graph. St000454The largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. 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. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000464The Schultz index of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St001118The acyclic chromatic index of a graph. 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 $eA$. St001545The second Elser number of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph. St000699The toughness times the least common multiple of 1,. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001060The distinguishing index of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.