Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 2
[[.,.],.] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => 6
[.,[[.,.],.]] => [2,3,1] => 4
[[.,.],[.,.]] => [3,1,2] => 5
[[.,[.,.]],.] => [2,1,3] => 3
[[[.,.],.],.] => [1,2,3] => 1
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 24
[.,[.,[[.,.],.]]] => [3,4,2,1] => 18
[.,[[.,.],[.,.]]] => [4,2,3,1] => 22
[.,[[.,[.,.]],.]] => [3,2,4,1] => 16
[.,[[[.,.],.],.]] => [2,3,4,1] => 10
[[.,.],[.,[.,.]]] => [4,3,1,2] => 23
[[.,.],[[.,.],.]] => [3,4,1,2] => 17
[[.,[.,.]],[.,.]] => [4,2,1,3] => 21
[[[.,.],.],[.,.]] => [4,1,2,3] => 19
[[.,[.,[.,.]]],.] => [3,2,1,4] => 15
[[.,[[.,.],.]],.] => [2,3,1,4] => 9
[[[.,.],[.,.]],.] => [3,1,2,4] => 13
[[[.,[.,.]],.],.] => [2,1,3,4] => 7
[[[[.,.],.],.],.] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 120
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 96
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 114
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 90
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 66
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 118
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 94
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 112
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 106
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 88
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 64
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 82
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 58
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 34
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 119
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 95
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 113
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 89
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 65
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 117
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 93
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 115
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 91
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 111
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 105
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 109
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 103
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 97
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 87
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 63
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 81
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 57
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 33
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 85
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 61
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 79
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 73
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 55
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 31
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 49
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 25
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 720
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 600
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 696
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 576
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 456
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 714
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 594
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 690
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 666
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 570
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 450
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 546
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 426
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 306
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 718
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 598
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 694
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 574
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 454
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 712
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 592
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 706
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 586
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 688
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 664
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 682
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 658
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 634
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 568
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 448
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 544
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 424
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 304
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 562
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 442
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 538
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 514
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 418
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 298
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 394
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 274
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 154
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 719
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 599
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 695
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 575
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 455
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 713
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 593
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 689
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 665
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 569
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 449
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 545
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 425
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 305
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 717
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 597
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 693
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 573
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 453
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 715
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 595
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 691
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 571
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 451
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 711
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 591
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 705
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 585
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 709
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 589
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 703
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 583
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 697
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 577
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 687
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 663
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 681
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 657
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 633
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 685
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 661
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 679
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 673
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 655
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 631
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 649
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 625
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 601
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 567
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 447
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 543
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 423
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 303
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 561
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 441
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 537
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 513
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 417
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 297
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 393
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 273
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 153
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 565
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 445
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 541
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 421
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 301
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 559
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 439
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 553
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 433
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 535
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 511
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 529
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 505
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 481
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 415
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 295
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 391
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 271
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 151
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 409
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 289
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 385
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 361
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 265
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 145
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 241
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 121
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 1
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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
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.