Your data matches 139 different statistics following compositions of up to 3 maps.
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Matching statistic: St000543
(load all 531 compositions to match this statistic)
Mapping
Mp00130: Permutations —descent tops⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 1 = 0 + 1
[2,1] => 1 => 1 = 0 + 1
[1,2,3] => 00 => 1 = 0 + 1
[1,3,2] => 01 => 2 = 1 + 1
[2,1,3] => 10 => 2 = 1 + 1
[2,3,1] => 01 => 2 = 1 + 1
[3,1,2] => 01 => 2 = 1 + 1
[3,2,1] => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 3 = 2 + 1
[1,3,2,4] => 010 => 3 = 2 + 1
[1,3,4,2] => 001 => 3 = 2 + 1
[1,4,2,3] => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 3 = 2 + 1
[2,1,4,3] => 101 => 3 = 2 + 1
[2,3,1,4] => 010 => 3 = 2 + 1
[2,3,4,1] => 001 => 3 = 2 + 1
[2,4,1,3] => 001 => 3 = 2 + 1
[2,4,3,1] => 011 => 3 = 2 + 1
[3,1,2,4] => 010 => 3 = 2 + 1
[3,1,4,2] => 011 => 3 = 2 + 1
[3,2,1,4] => 110 => 3 = 2 + 1
[3,2,4,1] => 011 => 3 = 2 + 1
[3,4,1,2] => 001 => 3 = 2 + 1
[3,4,2,1] => 101 => 3 = 2 + 1
[4,1,2,3] => 001 => 3 = 2 + 1
[4,1,3,2] => 011 => 3 = 2 + 1
[4,2,1,3] => 101 => 3 = 2 + 1
[4,2,3,1] => 011 => 3 = 2 + 1
[4,3,1,2] => 011 => 3 = 2 + 1
[4,3,2,1] => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 1 = 0 + 1
[1,2,3,4,5,6,7] => 000000 => 1 = 0 + 1
[7,6,5,4,3,2,1] => 111111 => 1 = 0 + 1
Description
The size of the conjugacy class of a binary word. Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
Matching statistic: St000293
(load all 5 compositions to match this statistic)
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 0
[2,1] => 1 => 1 => 0
[1,2,3] => 00 => 00 => 0
[1,3,2] => 01 => 10 => 1
[2,1,3] => 10 => 10 => 1
[2,3,1] => 10 => 10 => 1
[3,1,2] => 10 => 10 => 1
[3,2,1] => 11 => 11 => 0
[1,2,3,4] => 000 => 000 => 0
[1,2,4,3] => 001 => 100 => 2
[1,3,2,4] => 010 => 100 => 2
[1,3,4,2] => 010 => 100 => 2
[1,4,2,3] => 010 => 100 => 2
[1,4,3,2] => 011 => 110 => 2
[2,1,3,4] => 100 => 100 => 2
[2,1,4,3] => 101 => 110 => 2
[2,3,1,4] => 100 => 100 => 2
[2,3,4,1] => 100 => 100 => 2
[2,4,1,3] => 100 => 100 => 2
[2,4,3,1] => 101 => 110 => 2
[3,1,2,4] => 100 => 100 => 2
[3,1,4,2] => 110 => 110 => 2
[3,2,1,4] => 110 => 110 => 2
[3,2,4,1] => 110 => 110 => 2
[3,4,1,2] => 100 => 100 => 2
[3,4,2,1] => 110 => 110 => 2
[4,1,2,3] => 100 => 100 => 2
[4,1,3,2] => 110 => 110 => 2
[4,2,1,3] => 110 => 110 => 2
[4,2,3,1] => 110 => 110 => 2
[4,3,1,2] => 101 => 110 => 2
[4,3,2,1] => 111 => 111 => 0
[1,2,3,4,5] => 0000 => 0000 => 0
[5,4,3,2,1] => 1111 => 1111 => 0
[1,2,3,4,5,6] => 00000 => 00000 => 0
[6,5,4,3,2,1] => 11111 => 11111 => 0
[1,2,3,4,5,6,7] => 000000 => 000000 => 0
[7,6,5,4,3,2,1] => 111111 => 111111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St000682
(load all 7 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 0
[2,1] => 1 => 1 => 0
[1,2,3] => 00 => 00 => 0
[1,3,2] => 01 => 01 => 1
[2,1,3] => 10 => 01 => 1
[2,3,1] => 01 => 01 => 1
[3,1,2] => 10 => 01 => 1
[3,2,1] => 11 => 11 => 0
[1,2,3,4] => 000 => 000 => 0
[1,2,4,3] => 001 => 001 => 2
[1,3,2,4] => 010 => 001 => 2
[1,3,4,2] => 001 => 001 => 2
[1,4,2,3] => 010 => 001 => 2
[1,4,3,2] => 011 => 011 => 2
[2,1,3,4] => 100 => 001 => 2
[2,1,4,3] => 101 => 011 => 2
[2,3,1,4] => 010 => 001 => 2
[2,3,4,1] => 001 => 001 => 2
[2,4,1,3] => 010 => 001 => 2
[2,4,3,1] => 011 => 011 => 2
[3,1,2,4] => 100 => 001 => 2
[3,1,4,2] => 101 => 011 => 2
[3,2,1,4] => 110 => 011 => 2
[3,2,4,1] => 101 => 011 => 2
[3,4,1,2] => 010 => 001 => 2
[3,4,2,1] => 011 => 011 => 2
[4,1,2,3] => 100 => 001 => 2
[4,1,3,2] => 101 => 011 => 2
[4,2,1,3] => 110 => 011 => 2
[4,2,3,1] => 101 => 011 => 2
[4,3,1,2] => 110 => 011 => 2
[4,3,2,1] => 111 => 111 => 0
[1,2,3,4,5] => 0000 => 0000 => 0
[5,4,3,2,1] => 1111 => 1111 => 0
[1,2,3,4,5,6] => 00000 => 00000 => 0
[6,5,4,3,2,1] => 11111 => 11111 => 0
[1,2,3,4,5,6,7] => 000000 => 000000 => 0
[7,6,5,4,3,2,1] => 111111 => 111111 => 0
Description
The Grundy value of Welter's game on a binary word. Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
Matching statistic: St001436
(load all 5 compositions to match this statistic)
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St001436: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 0
[2,1] => 1 => 1 => 0
[1,2,3] => 00 => 00 => 0
[1,3,2] => 01 => 10 => 1
[2,1,3] => 10 => 10 => 1
[2,3,1] => 10 => 10 => 1
[3,1,2] => 10 => 10 => 1
[3,2,1] => 11 => 11 => 0
[1,2,3,4] => 000 => 000 => 0
[1,2,4,3] => 001 => 100 => 2
[1,3,2,4] => 010 => 100 => 2
[1,3,4,2] => 010 => 100 => 2
[1,4,2,3] => 010 => 100 => 2
[1,4,3,2] => 011 => 110 => 2
[2,1,3,4] => 100 => 100 => 2
[2,1,4,3] => 101 => 110 => 2
[2,3,1,4] => 100 => 100 => 2
[2,3,4,1] => 100 => 100 => 2
[2,4,1,3] => 100 => 100 => 2
[2,4,3,1] => 101 => 110 => 2
[3,1,2,4] => 100 => 100 => 2
[3,1,4,2] => 110 => 110 => 2
[3,2,1,4] => 110 => 110 => 2
[3,2,4,1] => 110 => 110 => 2
[3,4,1,2] => 100 => 100 => 2
[3,4,2,1] => 110 => 110 => 2
[4,1,2,3] => 100 => 100 => 2
[4,1,3,2] => 110 => 110 => 2
[4,2,1,3] => 110 => 110 => 2
[4,2,3,1] => 110 => 110 => 2
[4,3,1,2] => 101 => 110 => 2
[4,3,2,1] => 111 => 111 => 0
[1,2,3,4,5] => 0000 => 0000 => 0
[5,4,3,2,1] => 1111 => 1111 => 0
[1,2,3,4,5,6] => 00000 => 00000 => 0
[6,5,4,3,2,1] => 11111 => 11111 => 0
[1,2,3,4,5,6,7] => 000000 => 000000 => 0
[7,6,5,4,3,2,1] => 111111 => 111111 => 0
Description
The index of a given binary word in the lex-order among all its cyclic shifts.
Matching statistic: St001633
(load all 48 compositions to match this statistic)
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => ([(0,1)],2)
 => 0
[2,1] => 1 => ([(0,1)],2)
 => 0
[1,2,3] => 00 => ([(0,2),(2,1)],3)
 => 0
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
 => 0
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,3,4,5,6,7] => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
[7,6,5,4,3,2,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000047
(load all 12 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
 => [2] => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => [1,1] => 1 = 0 + 1
[1,2,3] => ([],3)
 => [3] => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => [2,1] => 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
 => [2,1] => 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,1] => 2 = 1 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,1] => 2 = 1 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => ([],4)
 => [4] => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => [3,1] => 3 = 2 + 1
[1,3,2,4] => ([(2,3)],4)
 => [3,1] => 3 = 2 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
 => [3,1] => 3 = 2 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => 3 = 2 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 3 = 2 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 3 = 2 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 3 = 2 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => ([],5)
 => [5] => 1 = 0 + 1
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5,6] => ([],6)
 => [6] => 1 = 0 + 1
[6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5,6,7] => ([],7)
 => [7] => 1 = 0 + 1
[7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,1,1] => 1 = 0 + 1
Description
The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000529
(load all 12 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000529: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 1 = 0 + 1
[2,1] => 1 => 1 => 1 = 0 + 1
[1,2,3] => 00 => 00 => 1 = 0 + 1
[1,3,2] => 01 => 01 => 2 = 1 + 1
[2,1,3] => 10 => 01 => 2 = 1 + 1
[2,3,1] => 01 => 01 => 2 = 1 + 1
[3,1,2] => 10 => 01 => 2 = 1 + 1
[3,2,1] => 11 => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 001 => 3 = 2 + 1
[1,3,2,4] => 010 => 001 => 3 = 2 + 1
[1,3,4,2] => 001 => 001 => 3 = 2 + 1
[1,4,2,3] => 010 => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 001 => 3 = 2 + 1
[2,1,4,3] => 101 => 011 => 3 = 2 + 1
[2,3,1,4] => 010 => 001 => 3 = 2 + 1
[2,3,4,1] => 001 => 001 => 3 = 2 + 1
[2,4,1,3] => 010 => 001 => 3 = 2 + 1
[2,4,3,1] => 011 => 011 => 3 = 2 + 1
[3,1,2,4] => 100 => 001 => 3 = 2 + 1
[3,1,4,2] => 101 => 011 => 3 = 2 + 1
[3,2,1,4] => 110 => 011 => 3 = 2 + 1
[3,2,4,1] => 101 => 011 => 3 = 2 + 1
[3,4,1,2] => 010 => 001 => 3 = 2 + 1
[3,4,2,1] => 011 => 011 => 3 = 2 + 1
[4,1,2,3] => 100 => 001 => 3 = 2 + 1
[4,1,3,2] => 101 => 011 => 3 = 2 + 1
[4,2,1,3] => 110 => 011 => 3 = 2 + 1
[4,2,3,1] => 101 => 011 => 3 = 2 + 1
[4,3,1,2] => 110 => 011 => 3 = 2 + 1
[4,3,2,1] => 111 => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 11111 => 1 = 0 + 1
[1,2,3,4,5,6,7] => 000000 => 000000 => 1 = 0 + 1
[7,6,5,4,3,2,1] => 111111 => 111111 => 1 = 0 + 1
Description
The number of permutations whose descent word is the given binary word. This is the sizes of the preimages of the map [[Mp00109]].
Matching statistic: St000626
(load all 12 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 1 = 0 + 1
[2,1] => 1 => 1 => 1 = 0 + 1
[1,2,3] => 00 => 00 => 1 = 0 + 1
[1,3,2] => 01 => 01 => 2 = 1 + 1
[2,1,3] => 10 => 01 => 2 = 1 + 1
[2,3,1] => 01 => 01 => 2 = 1 + 1
[3,1,2] => 10 => 01 => 2 = 1 + 1
[3,2,1] => 11 => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 001 => 3 = 2 + 1
[1,3,2,4] => 010 => 001 => 3 = 2 + 1
[1,3,4,2] => 001 => 001 => 3 = 2 + 1
[1,4,2,3] => 010 => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 001 => 3 = 2 + 1
[2,1,4,3] => 101 => 011 => 3 = 2 + 1
[2,3,1,4] => 010 => 001 => 3 = 2 + 1
[2,3,4,1] => 001 => 001 => 3 = 2 + 1
[2,4,1,3] => 010 => 001 => 3 = 2 + 1
[2,4,3,1] => 011 => 011 => 3 = 2 + 1
[3,1,2,4] => 100 => 001 => 3 = 2 + 1
[3,1,4,2] => 101 => 011 => 3 = 2 + 1
[3,2,1,4] => 110 => 011 => 3 = 2 + 1
[3,2,4,1] => 101 => 011 => 3 = 2 + 1
[3,4,1,2] => 010 => 001 => 3 = 2 + 1
[3,4,2,1] => 011 => 011 => 3 = 2 + 1
[4,1,2,3] => 100 => 001 => 3 = 2 + 1
[4,1,3,2] => 101 => 011 => 3 = 2 + 1
[4,2,1,3] => 110 => 011 => 3 = 2 + 1
[4,2,3,1] => 101 => 011 => 3 = 2 + 1
[4,3,1,2] => 110 => 011 => 3 = 2 + 1
[4,3,2,1] => 111 => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 11111 => 1 = 0 + 1
[1,2,3,4,5,6,7] => 000000 => 000000 => 1 = 0 + 1
[7,6,5,4,3,2,1] => 111111 => 111111 => 1 = 0 + 1
Description
The minimal period of a binary word. This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.
Matching statistic: St001313
(load all 7 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 1 = 0 + 1
[2,1] => 1 => 1 => 1 = 0 + 1
[1,2,3] => 00 => 00 => 1 = 0 + 1
[1,3,2] => 01 => 01 => 2 = 1 + 1
[2,1,3] => 10 => 01 => 2 = 1 + 1
[2,3,1] => 01 => 01 => 2 = 1 + 1
[3,1,2] => 10 => 01 => 2 = 1 + 1
[3,2,1] => 11 => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 000 => 1 = 0 + 1
[1,2,4,3] => 001 => 001 => 3 = 2 + 1
[1,3,2,4] => 010 => 001 => 3 = 2 + 1
[1,3,4,2] => 001 => 001 => 3 = 2 + 1
[1,4,2,3] => 010 => 001 => 3 = 2 + 1
[1,4,3,2] => 011 => 011 => 3 = 2 + 1
[2,1,3,4] => 100 => 001 => 3 = 2 + 1
[2,1,4,3] => 101 => 011 => 3 = 2 + 1
[2,3,1,4] => 010 => 001 => 3 = 2 + 1
[2,3,4,1] => 001 => 001 => 3 = 2 + 1
[2,4,1,3] => 010 => 001 => 3 = 2 + 1
[2,4,3,1] => 011 => 011 => 3 = 2 + 1
[3,1,2,4] => 100 => 001 => 3 = 2 + 1
[3,1,4,2] => 101 => 011 => 3 = 2 + 1
[3,2,1,4] => 110 => 011 => 3 = 2 + 1
[3,2,4,1] => 101 => 011 => 3 = 2 + 1
[3,4,1,2] => 010 => 001 => 3 = 2 + 1
[3,4,2,1] => 011 => 011 => 3 = 2 + 1
[4,1,2,3] => 100 => 001 => 3 = 2 + 1
[4,1,3,2] => 101 => 011 => 3 = 2 + 1
[4,2,1,3] => 110 => 011 => 3 = 2 + 1
[4,2,3,1] => 101 => 011 => 3 = 2 + 1
[4,3,1,2] => 110 => 011 => 3 = 2 + 1
[4,3,2,1] => 111 => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 0000 => 1 = 0 + 1
[5,4,3,2,1] => 1111 => 1111 => 1 = 0 + 1
[1,2,3,4,5,6] => 00000 => 00000 => 1 = 0 + 1
[6,5,4,3,2,1] => 11111 => 11111 => 1 = 0 + 1
[1,2,3,4,5,6,7] => 000000 => 000000 => 1 = 0 + 1
[7,6,5,4,3,2,1] => 111111 => 111111 => 1 = 0 + 1
Description
The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$. See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St000290
Mapping
Mp00130: Permutations —descent tops⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0 => 0 => 0
[2,1] => 1 => 1 => 1 => 0
[1,2,3] => 00 => 00 => 00 => 0
[1,3,2] => 01 => 01 => 10 => 1
[2,1,3] => 10 => 01 => 10 => 1
[2,3,1] => 01 => 01 => 10 => 1
[3,1,2] => 01 => 01 => 10 => 1
[3,2,1] => 11 => 11 => 11 => 0
[1,2,3,4] => 000 => 000 => 000 => 0
[1,2,4,3] => 001 => 001 => 010 => 2
[1,3,2,4] => 010 => 001 => 010 => 2
[1,3,4,2] => 001 => 001 => 010 => 2
[1,4,2,3] => 001 => 001 => 010 => 2
[1,4,3,2] => 011 => 011 => 110 => 2
[2,1,3,4] => 100 => 001 => 010 => 2
[2,1,4,3] => 101 => 011 => 110 => 2
[2,3,1,4] => 010 => 001 => 010 => 2
[2,3,4,1] => 001 => 001 => 010 => 2
[2,4,1,3] => 001 => 001 => 010 => 2
[2,4,3,1] => 011 => 011 => 110 => 2
[3,1,2,4] => 010 => 001 => 010 => 2
[3,1,4,2] => 011 => 011 => 110 => 2
[3,2,1,4] => 110 => 011 => 110 => 2
[3,2,4,1] => 011 => 011 => 110 => 2
[3,4,1,2] => 001 => 001 => 010 => 2
[3,4,2,1] => 101 => 011 => 110 => 2
[4,1,2,3] => 001 => 001 => 010 => 2
[4,1,3,2] => 011 => 011 => 110 => 2
[4,2,1,3] => 101 => 011 => 110 => 2
[4,2,3,1] => 011 => 011 => 110 => 2
[4,3,1,2] => 011 => 011 => 110 => 2
[4,3,2,1] => 111 => 111 => 111 => 0
[1,2,3,4,5] => 0000 => 0000 => 0000 => 0
[5,4,3,2,1] => 1111 => 1111 => 1111 => 0
[1,2,3,4,5,6] => 00000 => 00000 => 00000 => 0
[6,5,4,3,2,1] => 11111 => 11111 => 11111 => 0
[1,2,3,4,5,6,7] => 000000 => 000000 => 000000 => 0
[7,6,5,4,3,2,1] => 111111 => 111111 => 111111 => 0
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001280The number of parts of an integer partition that are at least two. St001485The modular major index of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St000003The number of standard Young tableaux of the partition. St000277The number of ribbon shaped standard tableaux. St000548The number of different non-empty partial sums of an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001958The degree of the polynomial interpolating the values of a permutation. St000387The matching number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000222The number of alignments in the permutation. St000362The size of a minimal vertex cover of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001535The number of cyclic alignments of a permutation. St000045The number of linear extensions of a binary tree. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001812The biclique partition number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001060The distinguishing index of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. St000264The girth of a graph, which is not a tree. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001651The Frankl number of a lattice. St001656The monophonic position number of a graph. St001654The monophonic hull number of a graph. St000768The number of peaks in an integer composition. St001470The cyclic holeyness of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001742The difference of the maximal and the minimal degree in a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000094The depth of an ordered tree. St001391The disjunction number of a graph. 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. St001964The interval resolution global dimension of a poset. 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. 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001118The acyclic chromatic index of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001488The number of corners of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1. 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. St000181The number of connected components of the Hasse diagram for the poset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. 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. 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. St001890The maximum magnitude of the Möbius function of a poset. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St001568The smallest positive integer that does not appear twice in the partition. St000718The largest Laplacian eigenvalue of a graph if it is integral.