A307797 Lexicographically earliest version of a self referencing "Kimberling shuffle" expulsion array sequence.

1, 2, 2, 3, 2, 4, 4, 5, 2, 2, 6, 2, 7, 8, 4, 4, 2, 6, 9, 2, 10, 4, 2, 2, 2, 11, 12, 4, 6, 11, 7, 6, 9, 9, 6, 2, 11, 11, 7, 13, 11, 2, 2, 7, 14, 2, 11, 9, 15, 16, 17, 17, 10, 7, 11, 2, 18, 11, 4, 7, 2, 12, 19, 18, 11, 4, 10, 11, 10, 6, 11, 13, 20, 21, 14, 6, 22, 23, 2, 24, 4, 15, 2, 2, 25, 20, 26, 12, 27, 7, 16, 28, 29, 30, 11, 31, 13, 29, 11

Offset: 1

Author

David James Sycamore and Lars Blomberg, Apr 29 2019

Keywords

• nonn

Comments

Start with this sequence, "shuffle" as in A007063 and the sequence reappears in the diagonal of the array. Terms are transformed from A307536 to this lexicofirst version by replacing the first and all subsequent occurrences of any term > all preceding terms by k+1, where k is the greatest (transformed) term seen so far. The records of this sequence is the natural numbers, A000027, starting point of the original Kimberling exclusion array.

Examples

			A307536(4)=4 > all preceding terms, the greatest of which is 2, so a(4)=3. Since 4 appears only once in A307536, 3 appears only once in this sequence.
A307536(21)=21 > all preceding terms, the greatest of which (in this sequence) is 9, so a(21)=10. Subsequent terms with the same value are a(53), a(67), a(69), ... because the corresponding terms (same indices) in A307536 all have value 21.
		

Links

• D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.
• C. Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, pp. 82-83.

Crossrefs

Cf. A007063, A307536, A006852, A000027.

Programs

PARI

A(z) = {x=z; y=z; xx=2*x-4; while (y<=xx, x--; xx-=2; if (bittest(y, 0)==1, y=x+((y+1)>>1), y=x-(y>>1))); return(x+y-1); } \\ A007063
B(z) = {a=z; n=1; while (a!=n, if (a2*n, a--, a=2*(a-n)-1); n++); return(a);} \\ A006852
addgroup(group, n, fixed, v) = {my(ok = 1, m=v[n]); while(ok, listput(group, m); if (m==n, ok=0; break); if (m > #v, ok=0; break); n = m; m = v[n];); group;}
makegroup(n, fixed, va, vb) = {my(group = List()); listput(group, n); group = addgroup(group, n, fixed, va); group = addgroup(group, n, fixed, vb); listsort(group, 1); Vec(group);}
setgroup(v, n, group) = {my(gmin = vecmin(group)); for (i=1, #group, if ((group[i] <= #v) && !v[n], v[n] = gmin);); v;}
lista() = {nn = 200; nout = 90; va = vector(nn, k, A(k)); vb = vector(nn, k, B(k)); vc = vector(nn); fixed = List(); for (n = 1, nn, if (va[n] == n, listput(fixed, n));); fixed = Vec(fixed); for (n=1, nn, group = makegroup(n, fixed, va, vb); vc = setgroup(vc, n, group);); vector(nout, k, vc[k]);} \\ A307536
earliest(v) = {my(m = Map(), val=1); for (i=1, #v, if (!mapisdefined(m, v[i]), mapput(m, v[i], val); val++);); apply(x->mapget(m, x), v);}
earliest(lista()) \\ Michel Marcus, Jun 14 2019