- FindStat

Identifier

Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000020: Permutations ⟶ ℤ

Values

[+] => [1] => 1

[-] => [1] => 1

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

[-,+] => [2,1] => 2

[+,-] => [1,2] => 1

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

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

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

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

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

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

[-,-,+] => [3,1,2] => 5

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

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

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

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

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

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

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

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

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

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

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

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

[-,+,+,+] => [2,3,4,1] => 10

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

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

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

[-,-,+,+] => [3,4,1,2] => 17

[-,+,-,+] => [2,4,1,3] => 11

[-,+,+,-] => [2,3,1,4] => 9

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

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

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

[-,-,-,+] => [4,1,2,3] => 19

[-,-,+,-] => [3,1,2,4] => 13

[-,+,-,-] => [2,1,3,4] => 7

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

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

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

[-,+,4,3] => [2,3,1,4] => 9

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

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

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

[-,3,2,+] => [2,4,1,3] => 11

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

[-,3,2,-] => [2,1,3,4] => 7

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

[-,3,4,2] => [2,1,3,4] => 7

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

[-,4,2,3] => [2,3,1,4] => 9

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

[-,4,+,2] => [3,2,1,4] => 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,-,1,+] => [1,4,3,2] => 6

[3,+,1,-] => [2,1,3,4] => 7

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

[3,+,4,1] => [2,1,3,4] => 7

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

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

[3,4,2,1] => [2,1,3,4] => 7

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

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

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

[4,+,1,3] => [2,1,3,4] => 7

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

[4,+,+,1] => [2,3,1,4] => 9

[4,-,+,1] => [3,1,4,2] => 14

[4,+,-,1] => [2,1,4,3] => 8

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

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

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

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

[-,+,+,+,+] => [2,3,4,5,1] => 34

[+,-,+,+,+] => [1,3,4,5,2] => 10

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

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

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

[-,-,+,+,+] => [3,4,5,1,2] => 65

[-,+,-,+,+] => [2,4,5,1,3] => 41

[-,+,+,-,+] => [2,3,5,1,4] => 35

[-,+,+,+,-] => [2,3,4,1,5] => 33

[+,-,-,+,+] => [1,4,5,2,3] => 17

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

[+,-,+,+,-] => [1,3,4,2,5] => 9

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$F_{1} = 2\ q$

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

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

$F_{4} = 16\ q + 7\ q^{2} + 9\ q^{3} + 3\ q^{4} + 4\ q^{5} + q^{6} + 7\ q^{7} + 4\ q^{8} + 4\ q^{9} + q^{10} + 2\ q^{11} + 3\ q^{13} + q^{14} + q^{15} + q^{17} + q^{19}$

$F_{5} = 32\ q + 15\ q^{2} + 21\ q^{3} + 7\ q^{4} + 11\ q^{5} + 4\ q^{6} + 21\ q^{7} + 12\ q^{8} + 12\ q^{9} + 3\ q^{10} + 7\ q^{11} + q^{12} + 12\ q^{13} + 4\ q^{14} + 6\ q^{15} + q^{16} + 4\ q^{17} + q^{18} + 5\ q^{19} + q^{20} + 2\ q^{21} + q^{23} + 15\ q^{25} + 8\ q^{26} + 12\ q^{27} + 4\ q^{28} + 7\ q^{29} + 3\ q^{30} + 11\ q^{31} + 7\ q^{32} + 5\ q^{33} + q^{34} + 3\ q^{35} + 7\ q^{37} + 3\ q^{38} + 2\ q^{39} + 2\ q^{41} + 3\ q^{43} + q^{44} + 7\ q^{49} + 4\ q^{50} + 4\ q^{51} + q^{52} + 3\ q^{53} + q^{54} + 4\ q^{55} + 3\ q^{56} + q^{57} + q^{59} + 4\ q^{61} + q^{62} + q^{63} + q^{65} + 2\ q^{67} + 3\ q^{73} + q^{74} + q^{75} + q^{77} + q^{79} + q^{80} + q^{85} + q^{91} + q^{97}$

$F_{6} = 64\ q + 31\ q^{2} + 45\ q^{3} + 15\ q^{4} + 26\ q^{5} + 11\ q^{6} + 49\ q^{7} + 28\ q^{8} + 28\ q^{9} + 7\ q^{10} + 18\ q^{11} + 4\ q^{12} + 33\ q^{13} + 11\ q^{14} + 19\ q^{15} + 4\ q^{16} + 11\ q^{17} + 4\ q^{18} + 16\ q^{19} + 5\ q^{20} + 9\ q^{21} + q^{22} + 5\ q^{23} + q^{24} + 45\ q^{25} + 24\ q^{26} + 36\ q^{27} + 12\ q^{28} + 22\ q^{29} + 10\ q^{30} + 33\ q^{31} + 21\ q^{32} + 15\ q^{33} + 3\ q^{34} + 10\ q^{35} + q^{36} + 24\ q^{37} + 10\ q^{38} + 9\ q^{39} + q^{40} + 7\ q^{41} + q^{42} + 12\ q^{43} + 5\ q^{44} + 3\ q^{45} + 2\ q^{47} + 28\ q^{49} + 16\ q^{50} + 16\ q^{51} + 4\ q^{52} + 12\ q^{53} + 4\ q^{54} + 19\ q^{55} + 13\ q^{56} + 7\ q^{57} + q^{58} + 6\ q^{59} + q^{60} + 16\ q^{61} + 4\ q^{62} + 7\ q^{63} + q^{64} + 4\ q^{65} + q^{66} + 9\ q^{67} + q^{68} + 3\ q^{69} + q^{71} + 15\ q^{73} + 5\ q^{74} + 7\ q^{75} + q^{76} + 5\ q^{77} + q^{78} + 9\ q^{79} + 5\ q^{80} + 2\ q^{81} + 2\ q^{83} + 7\ q^{85} + q^{86} + 2\ q^{87} + q^{89} + 5\ q^{91} + q^{92} + 2\ q^{93} + q^{95} + 6\ q^{97} + q^{98} + 2\ q^{99} + q^{101} + 3\ q^{103} + q^{104} + 2\ q^{109} + q^{115} + 31\ q^{121} + 16\ q^{122} + 24\ q^{123} + 8\ q^{124} + 15\ q^{125} + 7\ q^{126} + 28\ q^{127} + 16\ q^{128} + 16\ q^{129} + 4\ q^{130} + 11\ q^{131} + 3\ q^{132} + 21\ q^{133} + 7\ q^{134} + 13\ q^{135} + 3\ q^{136} + 7\ q^{137} + 3\ q^{138} + 11\ q^{139} + 4\ q^{140} + 7\ q^{141} + q^{142} + 4\ q^{143} + q^{144} + 26\ q^{145} + 15\ q^{146} + 22\ q^{147} + 8\ q^{148} + 14\ q^{149} + 7\ q^{150} + 16\ q^{151} + 11\ q^{152} + 6\ q^{153} + q^{154} + 4\ q^{155} + 12\ q^{157} + 6\ q^{158} + 3\ q^{159} + 3\ q^{161} + 6\ q^{163} + 3\ q^{164} + 18\ q^{169} + 11\ q^{170} + 12\ q^{171} + 3\ q^{172} + 9\ q^{173} + 3\ q^{174} + 9\ q^{175} + 7\ q^{176} + 2\ q^{177} + 2\ q^{179} + 9\ q^{181} + 3\ q^{182} + 2\ q^{183} + 2\ q^{185} + 5\ q^{187} + q^{188} + 10\ q^{193} + 4\ q^{194} + 6\ q^{195} + q^{196} + 4\ q^{197} + q^{198} + 3\ q^{199} + 3\ q^{200} + 3\ q^{205} + q^{206} + 3\ q^{211} + q^{212} + 4\ q^{217} + q^{218} + 2\ q^{219} + q^{221} + 15\ q^{241} + 8\ q^{242} + 12\ q^{243} + 4\ q^{244} + 8\ q^{245} + 4\ q^{246} + 11\ q^{247} + 7\ q^{248} + 5\ q^{249} + q^{250} + 4\ q^{251} + q^{252} + 10\ q^{253} + 4\ q^{254} + 5\ q^{255} + q^{256} + 3\ q^{257} + q^{258} + 6\ q^{259} + 3\ q^{260} + 3\ q^{261} + 2\ q^{263} + 11\ q^{265} + 7\ q^{266} + 10\ q^{267} + 4\ q^{268} + 7\ q^{269} + 4\ q^{270} + 5\ q^{271} + 4\ q^{272} + q^{273} + q^{275} + 5\ q^{277} + 3\ q^{278} + q^{279} + q^{281} + 3\ q^{283} + 2\ q^{284} + 11\ q^{289} + 7\ q^{290} + 5\ q^{291} + q^{292} + 4\ q^{293} + q^{294} + 5\ q^{295} + 4\ q^{296} + q^{297} + q^{299} + 5\ q^{301} + q^{302} + q^{303} + q^{305} + 3\ q^{307} + 7\ q^{313} + 3\ q^{314} + 2\ q^{315} + 2\ q^{317} + 2\ q^{319} + 2\ q^{320} + 2\ q^{325} + 2\ q^{331} + 3\ q^{337} + q^{338} + 7\ q^{361} + 4\ q^{362} + 4\ q^{363} + q^{364} + 3\ q^{365} + q^{366} + 4\ q^{367} + 3\ q^{368} + q^{369} + q^{371} + 4\ q^{373} + q^{374} + q^{375} + q^{377} + 3\ q^{379} + q^{380} + q^{381} + q^{383} + 4\ q^{385} + 3\ q^{386} + 4\ q^{387} + q^{388} + 3\ q^{389} + q^{390} + q^{391} + q^{392} + q^{397} + q^{398} + q^{403} + q^{404} + 4\ q^{409} + 3\ q^{410} + q^{411} + q^{413} + q^{415} + q^{416} + q^{421} + q^{427} + 4\ q^{433} + q^{434} + q^{435} + q^{437} + q^{439} + q^{440} + q^{445} + q^{451} + 2\ q^{457} + 3\ q^{481} + q^{482} + q^{483} + q^{485} + q^{487} + q^{488} + q^{493} + q^{499} + q^{505} + q^{506} + q^{507} + q^{509} + q^{529} + q^{530} + q^{553} + q^{577} + q^{601}$

Description

The rank of the permutation.
This is its position among all permutations of the same size ordered lexicographically.
This can be computed using the Lehmer code of a permutation:
$$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$
where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.

Map

lower permutation

Description

The lower bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.