- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001671: Permutations ⟶ ℤ

Values

0 => [2] => [1,1,0,0] => [1,2] => 0
1 => [1,1] => [1,0,1,0] => [2,1] => 1
00 => [3] => [1,1,1,0,0,0] => [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0] => [1,3,2] => 2
10 => [1,2] => [1,0,1,1,0,0] => [2,1,3] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 3
000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 5
100 => [1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 7
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 6
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 5
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 5
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 9
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => 12
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 11
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => 5
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 10
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => 9
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => 14
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 6
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 10
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 4
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 9
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 8
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 13
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 3
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 8
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 7
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 12
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 6
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 11
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 10
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 15
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => 6
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,2,3,4,6,5,7] => 5
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,2,3,4,6,7,5] => 11
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,2,3,5,4,6,7] => 4
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,2,3,5,4,7,6] => 10
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,2,3,5,6,4,7] => 9
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,2,3,5,6,7,4] => 15
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,2,4,3,5,6,7] => 3
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,2,4,3,5,7,6] => 9
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5,7] => 8
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,2,4,3,6,7,5] => 14
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,2,4,5,3,6,7] => 7
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,2,4,5,3,7,6] => 13
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,2,4,5,6,3,7] => 12
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,2,4,5,6,7,3] => 18
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,3,2,4,5,6,7] => 2
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,3,2,4,5,7,6] => 8
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2,4,6,5,7] => 7
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,3,2,4,6,7,5] => 13
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,5,4,6,7] => 6
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,7,6] => 12
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,5,6,4,7] => 11
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,5,6,7,4] => 17
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,3,4,2,5,6,7] => 5
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,3,4,2,5,7,6] => 11
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,3,4,2,6,5,7] => 10
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,3,4,2,6,7,5] => 16
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,3,4,5,2,6,7] => 9
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,3,4,5,2,7,6] => 15
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,3,4,5,6,2,7] => 14
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,7,2] => 20
 => [1] => [1,0] => [1] => 0

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$F_{1} = 1 + q$

$F_{2} = 1 + q + q^{2} + q^{3}$

$F_{3} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{6}$

$F_{4} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 2\ q^{6} + 2\ q^{7} + q^{8} + q^{9} + q^{10}$

$F_{5} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + 3\ q^{8} + 3\ q^{9} + 3\ q^{10} + 2\ q^{11} + 2\ q^{12} + q^{13} + q^{14} + q^{15}$

Description

Haglund's hag of a permutation.
Let

$edif$ be the sum of the differences of exceedence tops and bottoms, let

$\pi_E$ the subsequence of exceedence tops and let

$\pi_N$ be the subsequence of non-exceedence tops. Finally, let

$L$ be the number of pairs of indices

$k < i$ such that

$\pi_k \leq i < \pi_i$ .
Then

$hag(\pi) = edif + inv(\pi_E) - inv(\pi_N) + L$ , where

$inv$ denotes the number of inversions of a word.

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.

Map

bounce path

Description

The bounce path determined by an integer composition.