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Identifier

St000756: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 3

[2,1] => 1

[1,2,3] => 6

[1,3,2] => 3

[2,1,3] => 4

[2,3,1] => 3

[3,1,2] => 1

[3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => 15

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

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

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

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

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

[1,3,2,4,5] => 12

[1,3,2,5,4] => 7

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

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

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

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

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

[1,4,2,5,3] => 7

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

[1,4,3,5,2] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,1,4,5] => 12

[2,3,1,5,4] => 7

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

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

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

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

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

[2,4,1,5,3] => 7

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

[2,4,3,5,1] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,4,1,5,2] => 7

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

[3,4,2,5,1] => 7

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

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

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

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

>>> Load all 1201 entries. <<<

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$F_{1} = q$

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

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

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

$F_{5} = 24\ q + 24\ q^{3} + 12\ q^{4} + 8\ q^{5} + 18\ q^{6} + 8\ q^{7} + 10\ q^{8} + 3\ q^{9} + 6\ q^{10} + 3\ q^{11} + 2\ q^{12} + q^{13} + q^{15}$

$F_{6} = 120\ q + 120\ q^{3} + 60\ q^{4} + 40\ q^{5} + 90\ q^{6} + 64\ q^{7} + 50\ q^{8} + 39\ q^{9} + 42\ q^{10} + 23\ q^{11} + 28\ q^{12} + 13\ q^{13} + 10\ q^{14} + 8\ q^{15} + 6\ q^{16} + 3\ q^{17} + 2\ q^{18} + q^{19} + q^{21}$

Description

The sum of the positions of the left to right maxima of a permutation.
The generating function for this statistic is

$\sum_{\pi\in\mathfrak S_n} q^{slrmax(pi)} = \prod_{k=1}^n (q^k+k-1),$
see [prop. 2.6., 1].

References

[1] Kortchemski, I. Asymptotic behavior of permutation records arXiv:0804.0446
[2] Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is k (1<=k<=n(n+1)/2). OEIS:A143946

Code

def statistic(pi):
    """
    sage: r=5; factor(sum(x^statistic(pi) for pi in Permutations(r)))
    (x^5 + 4)*(x^4 + 3)*(x^3 + 2)*(x^2 + 1)*x
    """
    pi = list(pi)
    i = 0
    res = 0
    for j in range(len(pi)):
        if pi[j] > i:
            i = pi[j] 
            res += j+1
    return res

Created

Apr 08, 2017 at 21:39 by Martin Rubey

Updated

May 03, 2019 at 12:57 by Henning Ulfarsson