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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.

%I A356324 #16 Sep 10 2022 07:36:13
%S A356324 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,
%T A356324 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,
%U A356324 0,0,4,0,4,0,0,0,0,0,0,0,0,0,3,3,4,0,0
%N 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.
%C A356324 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.
%C A356324 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).
%C A356324 The number of terms corresponding to the permutations of [n] which vanish is A003319(n), and the numbers of nonzero terms is A356291(n).
%e A356324 Rows give the terms corresponding to the permutations of [n].
%e A356324 [0] [0]
%e A356324 [1] [0]
%e A356324 [2] [1, 0]
%e A356324 [3] [1, 1, 2, 0, 0, 0]
%e A356324 [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]
%o A356324 (SageMath)
%o A356324 def FirstSplit(p) -> int:
%o A356324     n = p.size()
%o A356324     for i in (1..n-1):
%o A356324         ok = True
%o A356324         for j in (1..i):
%o A356324             if not ok: break
%o A356324             for k in (i + 1..n):
%o A356324                 if p(j) > p(k):
%o A356324                     ok = False
%o A356324                     break
%o A356324         if ok: return i
%o A356324     return 0
%o A356324 def A356324_row(n): return [FirstSplit(p) for p in Permutations(n)]
%o A356324 for n in range(6): print(A356324_row(n))
%Y A356324 Cf. A003319, A356291, A059438, A030298.
%K A356324 nonn,tabf
%O A356324 0,7
%A A356324 _Peter Luschny_, Aug 03 2022