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Your data matches 215 different statistics following compositions of up to 3 maps.
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Matching statistic: St001958
(load all 282 compositions to match this statistic)

Mapping

St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 2 = 1 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => 3 = 2 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => 3 = 2 + 1
[2,4,1,3] => 3 = 2 + 1
[2,4,3,1] => 3 = 2 + 1
[3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => 3 = 2 + 1
[3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => 3 = 2 + 1
[3,4,1,2] => 3 = 2 + 1
[3,4,2,1] => 3 = 2 + 1
[4,1,2,3] => 3 = 2 + 1
[4,1,3,2] => 3 = 2 + 1
[4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => 3 = 2 + 1
[4,3,1,2] => 3 = 2 + 1
[4,3,2,1] => 1 = 0 + 1

Description

The degree of the polynomial interpolating the values of a permutation. Given a permutation

$\pi\in\mathfrak S_n$ there is a polynomial

$p$ of minimal degree such that

$p(n)=\pi(n)$ for

$n\in\{1,\dots,n\}$ . This statistic records the degree of

$p$ .

Matching statistic: St000543
(load all 150 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

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 2 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

Description

The number of inversions of a binary word.

Matching statistic: St000682
(load all 2 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

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: St000848
(load all 6 compositions to match this statistic)

Mapping

Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000848: 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

Description

The balance constant multiplied with the number of linear extensions of a poset. A pair of elements

$x,y$ of a poset is

$\alpha$ -balanced if the proportion

$P(x,y)$ of linear extensions where

$x$ comes before

$y$ is between

$\alpha$ and

$1-\alpha$ . The balance constant of a poset is

$\max\min(P(x,y), P(y,x)).$ Kislitsyn [1] conjectured that every poset which is not a chain is

$1/3$ -balanced. Brightwell, Felsner and Trotter [2] show that it is at least

$(1-\sqrt 5)/10$ -balanced. Olson and Sagan [3] exhibit various posets that are

$1/2$ -balanced.

Matching statistic: St000849
(load all 8 compositions to match this statistic)

Mapping

Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000849: 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

Description

The number of 1/3-balanced pairs in a poset. A pair of elements

$x,y$ of a poset is

$\alpha$ -balanced if the proportion of linear extensions where

$x$ comes before

$y$ is between

$\alpha$ and

$1-\alpha$ . Kislitsyn [1] conjectured that every poset which is not a chain has a

$1/3$ -balanced pair. Brightwell, Felsner and Trotter [2] show that at least a

$(1-\sqrt 5)/10$ -balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a

$1/3$ -balanced pair.

Matching statistic: St001436
(load all 2 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

Description

The index of a given binary word in the lex-order among all its cyclic shifts.

Matching statistic: St001633
(load all 11 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

Description

The number of simple modules with projective dimension two in the incidence algebra of the poset.

Matching statistic: St000047
(load all 6 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

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 6 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

Description

The number of permutations whose descent word is the given binary word. This is the sizes of the preimages of the map [[Mp00109]].

The following 205 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000626The minimal period of a binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St000089The absolute variation of a composition. St000222The number of alignments in the permutation. St000290The major index of a binary word. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001280The number of parts of an integer partition that are at least two. St001306The number of induced paths on four vertices in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001485The modular major index of a binary word. St001535The number of cyclic alignments of a permutation. St001557The number of inversions of the second entry of a permutation. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. 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. St000075The orbit size of a standard tableau under promotion. St000209Maximum difference of elements in cycles. 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. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001282The number of graphs with the same chromatic polynomial. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St001760The number of prefix or suffix reversals needed to sort a permutation. 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. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001814The number of partitions interlacing the given partition. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000387The matching number of a graph. 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. St000362The size of a minimal vertex cover of a graph. St000045The number of linear extensions of a binary tree. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001812The biclique partition number of a graph. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St001645The pebbling number of a connected 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. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000454The largest eigenvalue of a graph if it is integral. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St001669The number of single rises in a Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000717The number of ordinal summands of a poset. St001623The number of doubly irreducible elements of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St001644The dimension of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000552The number of cut vertices of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001793The difference between the clique number and the chromatic number of a graph. St000785The number of distinct colouring schemes of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000098The chromatic number of a graph. St000422The energy of a graph, if it is integral. St000636The hull number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001875The number of simple modules with projective dimension at most 1. St001654The monophonic hull number of a graph. St000080The rank of the poset. St000166The depth minus 1 of an ordered tree. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000094The depth of an ordered tree. St000367The number of simsun double descents of a permutation. St000649The number of 3-excedences of a permutation. St000768The number of peaks in an integer composition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001742The difference of the maximal and the minimal degree in a graph. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000899The maximal number of repetitions of an integer composition. St000902 The minimal number of repetitions of an integer composition. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . 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. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group 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. St001395The number of strictly unfriendly partitions of a graph. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001935The number of ascents in a parking function. 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. St000273The domination number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. 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. St000916The packing number of a graph. St001114The number of odd descents of a permutation. 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

$eA$ . St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001339The irredundance number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001642The Prague dimension of a graph. St001667The maximal size of a pair of weak twins for a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001734The lettericity of a graph. St001928The number of non-overlapping descents in a permutation. St000553The number of blocks of a graph. St000906The length of the shortest maximal chain in a poset. St001391The disjunction number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St000550The number of modular elements of a lattice. St000735The last entry on the main diagonal of a standard tableau. St000915The Ore degree of a graph. St000741The Colin de Verdière graph invariant. St000455The second largest eigenvalue of a graph if it is integral. 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

$eA$ . St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001964The interval resolution global dimension of a poset. St001487The number of inner corners of a skew partition. 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. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices 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

$K[x]/(x^n)$ . St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St000181The number of connected components of the Hasse diagram for the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St000307The number of rowmotion orbits of a poset. 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.