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

Author

Peter Luschny, Aug 03 2022

Keywords

• nonn
• tabf

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).

Examples

			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]
		

Crossrefs

Cf. A003319, A356291, A059438, A030298.

Programs

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))