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
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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].
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
[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
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