- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000078: Permutations ⟶ ℤ

Values

[1] => [1] => 1

[1,2] => [1,2] => 1

[2,1] => [1,2] => 1

[1,2,3] => [1,2,3] => 1

[1,3,2] => [1,2,3] => 1

[2,1,3] => [1,2,3] => 1

[2,3,1] => [1,2,3] => 1

[3,1,2] => [1,3,2] => 2

[3,2,1] => [1,3,2] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,1,2,3] => [1,4,3,2] => 5

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

[4,2,1,3] => [1,4,3,2] => 5

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

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

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

[1,2,3,4,5] => [1,2,3,4,5] => 1

[1,2,3,5,4] => [1,2,3,4,5] => 1

[1,2,4,3,5] => [1,2,3,4,5] => 1

[1,2,4,5,3] => [1,2,3,4,5] => 1

[1,2,5,3,4] => [1,2,3,5,4] => 4

[1,2,5,4,3] => [1,2,3,5,4] => 4

[1,3,2,4,5] => [1,2,3,4,5] => 1

[1,3,2,5,4] => [1,2,3,4,5] => 1

[1,3,4,2,5] => [1,2,3,4,5] => 1

[1,3,4,5,2] => [1,2,3,4,5] => 1

[1,3,5,2,4] => [1,2,3,5,4] => 4

[1,3,5,4,2] => [1,2,3,5,4] => 4

[1,4,2,3,5] => [1,2,4,3,5] => 3

[1,4,2,5,3] => [1,2,4,5,3] => 6

[1,4,3,2,5] => [1,2,4,3,5] => 3

[1,4,3,5,2] => [1,2,4,5,3] => 6

[1,4,5,2,3] => [1,2,4,3,5] => 3

[1,4,5,3,2] => [1,2,4,3,5] => 3

[1,5,2,3,4] => [1,2,5,4,3] => 14

[1,5,2,4,3] => [1,2,5,3,4] => 6

[1,5,3,2,4] => [1,2,5,4,3] => 14

[1,5,3,4,2] => [1,2,5,3,4] => 6

[1,5,4,2,3] => [1,2,5,3,4] => 6

[1,5,4,3,2] => [1,2,5,3,4] => 6

[2,1,3,4,5] => [1,2,3,4,5] => 1

[2,1,3,5,4] => [1,2,3,4,5] => 1

[2,1,4,3,5] => [1,2,3,4,5] => 1

[2,1,4,5,3] => [1,2,3,4,5] => 1

[2,1,5,3,4] => [1,2,3,5,4] => 4

[2,1,5,4,3] => [1,2,3,5,4] => 4

[2,3,1,4,5] => [1,2,3,4,5] => 1

[2,3,1,5,4] => [1,2,3,4,5] => 1

[2,3,4,1,5] => [1,2,3,4,5] => 1

[2,3,4,5,1] => [1,2,3,4,5] => 1

[2,3,5,1,4] => [1,2,3,5,4] => 4

[2,3,5,4,1] => [1,2,3,5,4] => 4

[2,4,1,3,5] => [1,2,4,3,5] => 3

[2,4,1,5,3] => [1,2,4,5,3] => 6

[2,4,3,1,5] => [1,2,4,3,5] => 3

[2,4,3,5,1] => [1,2,4,5,3] => 6

[2,4,5,1,3] => [1,2,4,3,5] => 3

[2,4,5,3,1] => [1,2,4,3,5] => 3

[2,5,1,3,4] => [1,2,5,4,3] => 14

[2,5,1,4,3] => [1,2,5,3,4] => 6

[2,5,3,1,4] => [1,2,5,4,3] => 14

[2,5,3,4,1] => [1,2,5,3,4] => 6

[2,5,4,1,3] => [1,2,5,3,4] => 6

[2,5,4,3,1] => [1,2,5,3,4] => 6

[3,1,2,4,5] => [1,3,2,4,5] => 2

[3,1,2,5,4] => [1,3,2,4,5] => 2

[3,1,4,2,5] => [1,3,4,2,5] => 3

[3,1,4,5,2] => [1,3,4,5,2] => 4

[3,1,5,2,4] => [1,3,5,4,2] => 11

[3,1,5,4,2] => [1,3,5,2,4] => 8

[3,2,1,4,5] => [1,3,2,4,5] => 2

[3,2,1,5,4] => [1,3,2,4,5] => 2

[3,2,4,1,5] => [1,3,4,2,5] => 3

[3,2,4,5,1] => [1,3,4,5,2] => 4

[3,2,5,1,4] => [1,3,5,4,2] => 11

[3,2,5,4,1] => [1,3,5,2,4] => 8

[3,4,1,2,5] => [1,3,2,4,5] => 2

[3,4,1,5,2] => [1,3,2,4,5] => 2

[3,4,2,1,5] => [1,3,2,4,5] => 2

[3,4,2,5,1] => [1,3,2,4,5] => 2

[3,4,5,1,2] => [1,3,5,2,4] => 8

[3,4,5,2,1] => [1,3,5,2,4] => 8

[3,5,1,2,4] => [1,3,2,5,4] => 13

[3,5,1,4,2] => [1,3,2,5,4] => 13

>>> Load all 5914 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 8\ q + 4\ q^{2} + 10\ q^{3} + 2\ q^{5}$

$F_{5} = 16\ q + 8\ q^{2} + 20\ q^{3} + 18\ q^{4} + 4\ q^{5} + 16\ q^{6} + 6\ q^{7} + 8\ q^{8} + 6\ q^{9} + 2\ q^{10} + 6\ q^{11} + 4\ q^{13} + 6\ q^{14}$

$F_{6} = 32\ q + 16\ q^{2} + 40\ q^{3} + 36\ q^{4} + 42\ q^{5} + 32\ q^{6} + 12\ q^{7} + 16\ q^{8} + 22\ q^{9} + 60\ q^{10} + 12\ q^{11} + 10\ q^{12} + 14\ q^{13} + 32\ q^{14} + 44\ q^{15} + 8\ q^{16} + 2\ q^{17} + 20\ q^{19} + 36\ q^{20} + 8\ q^{21} + 6\ q^{22} + 14\ q^{23} + 4\ q^{24} + 12\ q^{25} + 32\ q^{26} + 6\ q^{27} + 6\ q^{28} + 8\ q^{29} + 8\ q^{30} + 6\ q^{31} + 22\ q^{32} + 12\ q^{34} + 12\ q^{35} + 6\ q^{37} + 4\ q^{38} + 20\ q^{40} + 2\ q^{42} + 2\ q^{43} + 8\ q^{44} + 4\ q^{46} + 6\ q^{53} + 6\ q^{58} + 4\ q^{67} + 8\ q^{74} + 4\ q^{84} + 4\ q^{121}$

$F_{7} = 64\ q + 32\ q^{2} + 80\ q^{3} + 72\ q^{4} + 84\ q^{5} + 130\ q^{6} + 24\ q^{7} + 32\ q^{8} + 44\ q^{9} + 120\ q^{10} + 42\ q^{11} + 20\ q^{12} + 28\ q^{13} + 64\ q^{14} + 210\ q^{15} + 26\ q^{16} + 38\ q^{17} + 18\ q^{18} + 40\ q^{19} + 138\ q^{20} + 22\ q^{21} + 12\ q^{22} + 46\ q^{23} + 78\ q^{24} + 36\ q^{25} + 66\ q^{26} + 32\ q^{27} + 22\ q^{28} + 58\ q^{29} + 38\ q^{30} + 24\ q^{31} + 78\ q^{32} + 16\ q^{33} + 58\ q^{34} + 32\ q^{35} + 32\ q^{36} + 22\ q^{37} + 8\ q^{38} + 42\ q^{39} + 124\ q^{40} + 6\ q^{41} + 24\ q^{42} + 34\ q^{43} + 24\ q^{44} + 90\ q^{45} + 24\ q^{46} + 44\ q^{47} + 38\ q^{48} + 82\ q^{50} + 12\ q^{51} + 24\ q^{53} + 70\ q^{54} + 58\ q^{55} + 16\ q^{56} + 8\ q^{57} + 12\ q^{58} + 44\ q^{60} + 20\ q^{61} + 26\ q^{62} + 18\ q^{63} + 16\ q^{64} + 46\ q^{65} + 46\ q^{66} + 8\ q^{67} + 2\ q^{68} + 46\ q^{69} + 38\ q^{70} + 36\ q^{71} + 30\ q^{72} + 12\ q^{73} + 42\ q^{74} + 74\ q^{75} + 26\ q^{76} + 26\ q^{77} + 8\ q^{78} + 20\ q^{79} + 32\ q^{80} + 16\ q^{81} + 8\ q^{82} + 46\ q^{84} + 58\ q^{85} + 20\ q^{87} + 36\ q^{89} + 32\ q^{90} + 6\ q^{93} + 14\ q^{94} + 54\ q^{95} + 6\ q^{96} + 16\ q^{97} + 6\ q^{98} + 10\ q^{100} + 8\ q^{101} + 12\ q^{102} + 14\ q^{104} + 18\ q^{105} + 2\ q^{107} + 6\ q^{108} + 26\ q^{109} + 28\ q^{110} + 6\ q^{114} + 36\ q^{115} + 16\ q^{117} + 12\ q^{118} + 40\ q^{120} + 8\ q^{121} + 16\ q^{122} + 30\ q^{124} + 44\ q^{125} + 14\ q^{126} + 32\ q^{127} + 12\ q^{129} + 20\ q^{130} + 22\ q^{131} + 6\ q^{132} + 12\ q^{135} + 8\ q^{138} + 8\ q^{139} + 12\ q^{142} + 16\ q^{144} + 20\ q^{145} + 8\ q^{146} + 4\ q^{149} + 12\ q^{150} + 16\ q^{151} + 10\ q^{152} + 24\ q^{153} + 20\ q^{154} + 16\ q^{155} + 6\ q^{159} + 26\ q^{160} + 18\ q^{162} + 6\ q^{163} + 18\ q^{165} + 30\ q^{170} + 22\ q^{172} + 6\ q^{180} + 18\ q^{183} + 12\ q^{184} + 18\ q^{185} + 12\ q^{190} + 6\ q^{191} + 6\ q^{193} + 22\ q^{195} + 20\ q^{198} + 8\ q^{200} + 12\ q^{202} + 28\ q^{205} + 12\ q^{208} + 8\ q^{210} + 6\ q^{211} + 12\ q^{212} + 12\ q^{215} + 6\ q^{216} + 12\ q^{221} + 8\ q^{224} + 2\ q^{225} + 12\ q^{227} + 12\ q^{230} + 8\ q^{234} + 12\ q^{235} + 4\ q^{243} + 20\ q^{250} + 2\ q^{251} + 8\ q^{254} + 8\ q^{255} + 12\ q^{266} + 12\ q^{267} + 6\ q^{291} + 24\ q^{294} + 12\ q^{295} + 8\ q^{297} + 12\ q^{300} + 16\ q^{310} + 12\ q^{320} + 6\ q^{324} + 18\ q^{328} + 8\ q^{330} + 12\ q^{335} + 12\ q^{360} + 6\ q^{378} + 4\ q^{381} + 12\ q^{398} + 8\ q^{426} + 8\ q^{432} + 4\ q^{434} + 8\ q^{439} + 12\ q^{456} + 4\ q^{477} + 12\ q^{480} + 12\ q^{496} + 8\ q^{504} + 6\ q^{553} + 4\ q^{594} + 4\ q^{610} + 8\ q^{743} + 8\ q^{888} + 4\ q^{1186} + 4\ q^{1514}$

Description

The number of alternating sign matrices whose left key is the permutation.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.