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Matching statistic: St000742
Mapping
St000742: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
Description
The number of big ascents of a permutation after prepending zero. Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$. It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]). G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]]. [[St000646]] is the statistic without the border condition $\pi(0) = 0$.