You can use sage and www.findstat.org to find a candidate for a bijection as follows. First define the statistics you are interested in:

```
def A_num(s, k):
return len([1 for i,e in enumerate(s,1) if e-i==k])
def B_num(s, k):
return len([1 for e,f in zip(s, s[1:]) if f-e==k])
```

Then ask, what findstat knows about them:

```
sage: findstat("Permutations", lambda s: A_num(s, 2), depth=3)
0: (St000534: The number of 2-rises of a permutation., [Mp00066: inverse, Mp00087: inverse first fundamental transformation, Mp00064: reverse], 200)
sage: findstat("Permutations", lambda s: B_num(s, 2), depth=3)
0: (St000534: The number of 2-rises of a permutation., [], 200)
sage: findstat("Permutations", lambda s: A_num(s, 1), depth=3)
0: (St000237: The number of indices $i$ such that $\pi_i=i+1$., [], 200)
1: (St000214: The number of adjacencies (or small descents) of a permutation., [Mp00066: inverse, Mp00087: inverse first fundamental transformation], 200)
2: (St000441: The number of successions (or small ascents) of a permutation., [Mp00066: inverse, Mp00087: inverse first fundamental transformation, Mp00064: reverse], 200)
sage: findstat("Permutations", lambda s: B_num(s, 1), depth=3)
0: (St000441: The number of successions (or small ascents) of a permutation., [], 200)
1: (St000214: The number of adjacencies (or small descents) of a permutation., [Mp00064: reverse], 200)
2: (St000237: The number of indices $i$ such that $\pi_i=i+1$., [Mp00064: reverse, Mp00086: first fundamental transformation, Mp00066: inverse], 200)
```

So, this suggests that using the composition of the maps http://www.findstat.org/MapsDatabase/Mp00066,
http://www.findstat.org/MapsDatabase/Mp00087
and
http://www.findstat.org/MapsDatabase/Mp00064
might be a good idea. No guarantee, of course.

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