cp's OEIS Frontend

A352513 Number of nonfixed points in the n-th composition in standard order.

%I A352513 #9 Apr 01 2022 23:52:32
%S A352513 0,0,1,1,1,2,0,2,1,2,1,3,1,1,2,3,1,2,1,3,2,2,3,4,1,2,1,2,1,3,3,4,1,2,
%T A352513 1,3,2,2,3,4,2,3,2,3,2,4,4,5,1,2,2,3,0,2,2,3,2,2,3,4,3,4,4,5,1,2,1,3,
%U A352513 2,2,3,4,2,3,2,3,2,4,4,5,2,3,3,4,1,3,3
%N A352513 Number of nonfixed points in the n-th composition in standard order.
%C A352513 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.
%C A352513 A nonfixed point in a composition c is an index i such that c_i != i.
%F A352513 A000120(n) = A352512(n) + A352513(n).
%e A352513 The 169th composition in standard order is (2,2,3,1), with nonfixed points {1,4}, so a(169) = 2.
%t A352513 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A352513 pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
%t A352513 Table[pnq[stc[n]],{n,0,100}]
%Y A352513 The version counting permutations is A098825, fixed A008290.
%Y A352513 Fixed points are counted by A352512, triangle A238349, first A238351.
%Y A352513 The triangular version is A352523, first nontrivial column A352520.
%Y A352513 A011782 counts compositions.
%Y A352513 A352486 gives the nonfixed points of A122111, counted by A330644.
%Y A352513 A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
%Y A352513 A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.
%Y A352513 A352524 counts comps by strong excedances, first col A008930, stat A352516.
%Y A352513 A352525 counts comps by weak excedances, first col A177510, stat A352517.
%Y A352513 Cf. A000700, A088218, A088902, A114088, A115994, A238352, A350841.
%K A352513 nonn
%O A352513 0,6
%A A352513 _Gus Wiseman_, Mar 27 2022