Kendell-Mann numbers $M(n)$ ( see the sequence A000140 http://oeis.org/A000140 ) have the simple property: $M(n+1) \approx (n-1/2)M(n)$.

The property can be proved by different methods. For eg. The property of Kendall-Mann numbers

What I am looking for is to find out if a combinatorial proof exists?

For eg. Let us start: Suppose we look at all the permutations of $n-1$ in the maximal grouping, then at all the permutation of $n$ in that maximal grouping; is there any simple way in which each permutation in the first set gives rise to $n$ permutations in the second? Better yet, a simple way in which about half the $n-1$-permutations give rise to $n$ $n$-permutations each, and the other half give rise to $n+1$ $n$-permutations each?

Any hints are higly welcomed. I hope that the combinatorial proof will makes the reason for the simple property more transparent.

Kendell-Mann numbers $M(n)$ ( see the sequence A000140 http://oeis.org/A000140 ) have the simple property: $M(n+1) \approx (n-1/2)M(n)$.

The property can be proved by different methods. For eg. The property of Kendall-Mann numbers

What I am looking for is to find out if a combinatorial proof exists?

For eg. Let us start: Suppose we look at all the permutations of $n-1$ in the maximal grouping, then at all the permutation of $n$ in that maximal grouping; is there any simple way in which each permutation in the first set gives rise to $n$ permutations in the second? Better yet, a simple way in which about half the $n-1$-permutations give rise to $n$ $n$-permutations each, and the other half give rise to $n+1$ $n$-permutations each?

Any hints are higly welcomed. I hope that the combinatorial proof will makes the reason for the simple property more transparent.