Recursive transformation ENIPS for Catalan bijections has a well-defined inverse (see the definition & comments at
A122204). For all Catalan bijections in
A089840 that inverse produces a bijection which is itself in
A089840. This sequence gives the indices to those positions where each ("primitive", non-recursive bijection) of
A089840(n) occurs "atavistically" amongst the more complex recursive bijections in
A122204. I.e.
A122204(a(n)) =
A089840(n). Similarly, other "atavistic forms" resurface as:
A122287(a(n)) =
A122201(n),
A122286(a(n)) =
A122203(n) and
A122202(a(n)) =
A122284(n). See also comments at
A153833.
There exists similar atavistic index sequences computed for FORK (
A122201) and KROF (
A122202). Both start as 0,1654720,... (see
A129604). This implies that regardless of how complex recursive derivations from
A089840 one forms by repeatedly applying SPINE, ENIPS, FORK and/or KROF in some order (finite number of times), all the original primitive non-recursive elements of
A089840 will eventually appear at some positions.
Other known terms: a(12)=65167, a(13)=65178, a(14)=65236, a(15)=169, a(16)=65302, a(22)-a(44) = 1656351, 1656576, 1656777, 1656628, 1656704, 1659507, 1659538, 1659653, 1659798, 1659685, 1659830, 1660155, 1660582, 1660439, 1660476, 1660621, 1660196, 1661073, 1660930, 1660859, 1661004, 1661287, 1661360.
