Graceful permutations

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A graceful permutation of the integers 1 to $n$ is a permutation such that the differences of consecutive terms consists of the numbers 1 to $n-1$. For example, {1, 7, 2, 6, 3, 5, 4} is a graceful permutation of the numbers 1 to 7, since the differences are {6, 5, 4, 3, 2, 1} (the numbers 1 to 6 backwards).

There are $n!$ permutations of $n$ numbers, so the total number of graceful permutations must be less than this for all $n>2$. See A006967.

Graceful permutations are most often defined in terms of graph theory, sometimes in terms of a generic alphabet of letters.

• Michał Adamaszek, "Efficient enumeration of graceful permutations"