A356324 a(n) is the first split point of the permutation p if p is the n-th permutation (in lexicographic order (A030298 prepended by the empty permutation)), or zero if it has no split point.

0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 4, 0, 0

Offset

0,7

Comments

A permutation p in [n] (where n >= 0) is reducible if there exist an i in 1..n-1 such that for all j in the range 1..i and all k in the range i+1..n it is true that p(j) < p(k). (Note that a range a..b includes a and b.) If such an i exists we say that i splits the permutation p at i and that i is a split point of p.

The list of permutations starts with the empty permutation (), which has no split points. The first permutation which has a split point is (1, 2).

The number of terms corresponding to the permutations of [n] which vanish is A003319(n), and the numbers of nonzero terms is A356291(n).

Links

Table of n, a(n) for n=0..86.

Example

Rows give the terms corresponding to the permutations of [n].
[0] [0]
[1] [0]
[2] [1, 0]
[3] [1, 1, 2, 0, 0, 0]
[4] [1, 1, 1, 1, 1, 1, 2, 2, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Prog

(SageMath)
def FirstSplit(p) -> int:
    n = p.size()
    for i in (1..n-1):
        ok = True
        for j in (1..i):
            if not ok: break
            for k in (i + 1..n):
                if p(j) > p(k):
                    ok = False
                    break
        if ok: return i
    return 0
def A356324_row(n): return [FirstSplit(p) for p in Permutations(n)]
for n in range(6): print(A356324_row(n))

Crossrefs

Cf. A003319, A356291, A059438, A030298.

Sequence in context: A330268 A089310 A129753 * A307247 A147693 A070936

Adjacent sequences: A356321 A356322 A356323 * A356325 A356326 A356327

Keyword

nonn,tabf

Author

Peter Luschny, Aug 03 2022

Status

approved