%I #37 Apr 22 2021 03:42:43
%S 1,1,2,4,8,12,24,34,62,88,148,208,360,466,784,1082,1718,2278,3744,
%T 4902,7914,10486,16334,21728
%N Number of distinct sets { p(i) - p(j) : 1 <= i <= j <= n } where p ranges over all permutations of [n].
%C a(n) is even for n > 1.
%F a(n) < 2 + 74*3^(n-6).
%F a(n) <= 2*a(n-1) (conjectured).
%e a(1) = 1: [[0]].
%e a(2) = 2: [[-1, 0], [0, 1]].
%e a(3) = 4: [[-2, -1, 0], [-2, -1, 0, 1], [-1, 0, 1, 2], [0, 1, 2]].
%e a(4) = 8: [[-3, -2, -1, 0], [-3, -2, -1, 0, 1], [-3, -2, -1, 0, 1, 2], [-2, -1, 0, 1, 2, 3], [-2, -1, 0, 1, 3], [-3, -1, 0, 1, 2], [-1, 0, 1, 2, 3], [0, 1, 2, 3]].
%e a(5) = 12: [[-4, -3, -2, -1, 0], [-4, -3, -2, -1, 0, 1], [-4, -3, -2, -1, 0, 1, 2], [-4, -3, -2, -1, 0, 1, 2, 3], [-4, -3, -2, -1, 0, 1, 3], [-3, -2, -1, 0, 1, 2, 3, 4], [-3, -2, -1, 0, 1, 2, 4], [-4, -2, -1, 0, 1, 2, 3], [-2, -1, 0, 1, 2, 3, 4], [-3, -1, 0, 1, 2, 3, 4], [-1, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4]].
%p b:= proc(s) option remember; `if`(s={}, {{}}, {seq(map(x->
%p {seq(j-i, j=s)} union x, b(s minus {i}))[], i=s)})
%p end:
%p a:= n-> nops(b({$1..n})):
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Apr 15 2021
%o (Python)
%o def perm(pmt,begin,end):
%o global k
%o global a_n
%o if begin>=end:
%o a=[]
%o for x in range(1,len(pmt)):
%o for y in range(0,x+1):
%o a.append(pmt[y]-pmt[x])
%o new_list=[]
%o for j in a:
%o if j not in new_list:
%o new_list.append(j)
%o new_list.sort()
%o k.append(new_list)
%o m=[]
%o for ss in k:
%o if ss not in m:
%o m.append(ss)
%o k=m
%o a_n=len(m)
%o else:
%o i=begin
%o for num in range(begin,end):
%o pmt[num],pmt[i]=pmt[i],pmt[num]
%o perm(pmt,begin+1,end)
%o pmt[num],pmt[i]=pmt[i],pmt[num]
%o N=1
%o while True:
%o k=[]
%o a_n=0
%o pmt=[]
%o for p in range(0,N):
%o pmt.append(p+1)
%o perm(pmt,0,len(pmt))
%o print("a(",N,")=",a_n)
%o N=N+1
%o (Python)
%o from itertools import permutations
%o def a(n): return len(set(tuple(sorted(set(p[i] - p[j] for i in range(n) for j in range(i, n)))) for p in permutations(range(1, n+1))))
%o print([a(n) for n in range(10)]) # _Michael S. Branicky_, Apr 17 2021
%Y Cf. A000142.
%K nonn,more
%O 0,3
%A _Baohua Tian_, Apr 15 2021
%E a(11)-a(16) from _Alois P. Heinz_, Apr 15 2021
%E a(17)-a(23) from _Bert Dobbelaere_, Apr 21 2021