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A352516 Number of excedances (parts above the diagonal) of the n-th composition in standard order.

%I A352516 #5 Mar 31 2022 03:08:03
%S A352516 0,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,1,1,1,1,2,1,1,1,1,1,0,0,0,0,0,0,1,1,
%T A352516 1,1,2,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,
%U A352516 2,1,1,1,2,2,1,1,1,1,1,1,2,2,2,2,1,1,1
%N A352516 Number of excedances (parts above the diagonal) of the n-th composition in standard order.
%C A352516 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.
%H A352516 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A352516 The 5392th composition in standard order is (2,2,4,5), with excedances {1,3,4}, so a(5392) = 3.
%t A352516 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A352516 pd[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t A352516 Table[pd[stc[n]],{n,0,30}]
%Y A352516 Positions of first appearances are A104462.
%Y A352516 The opposite version is A352514, counted by A352521 (first column A219282).
%Y A352516 The weak opposite version is A352515, counted by A352522 (first A238874).
%Y A352516 The weak version is A352517, counted by A352525 (first column A177510).
%Y A352516 The triangle A352524 counts these compositions (first column A008930).
%Y A352516 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352516 A011782 counts compositions.
%Y A352516 A173018 counts permutations by number of excedances, weak A123125.
%Y A352516 A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352516 A352487 is the excedance set of A122111.
%Y A352516 A352523 counts comps by unfixed points, first A010054, rank stat A352513.
%Y A352516 Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.
%K A352516 nonn
%O A352516 0,21
%A A352516 _Gus Wiseman_, Mar 23 2022