cp's OEIS Frontend

A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

%I A352515 #7 Mar 31 2022 03:07:51
%S A352515 0,1,0,2,0,1,2,3,0,1,1,2,1,3,3,4,0,1,1,2,0,2,2,3,1,2,3,4,3,4,4,5,0,1,
%T A352515 1,2,0,2,2,3,0,1,2,3,2,3,3,4,1,2,2,3,3,4,4,5,2,4,4,5,4,5,5,6,0,1,1,2,
%U A352515 0,2,2,3,0,1,2,3,2,3,3,4,0,1,1,2,2,3,3
%N A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.
%C A352515 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 A352515 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A352515 The 89th composition in standard order is (2,1,3,1), with weak nonexcedances {2,3,4}, so a(89) = 3.
%t A352515 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A352515 paw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
%t A352515 Table[paw[stc[n]],{n,0,30}]
%Y A352515 Positions of first appearances are A000225.
%Y A352515 The strong version is A352514, counted by A352521 (first column A219282).
%Y A352515 The strong opposite version is A352516, counted by A352524 (first A008930).
%Y A352515 The opposite version is A352517, counted by A352525 (first column A177510).
%Y A352515 Triangle A352522 counts these comps (first col A238874), partitions A115994.
%Y A352515 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352515 A011782 counts compositions.
%Y A352515 A173018 counts permutations by number of excedances, weak A123125.
%Y A352515 A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352515 A352488 is the weak nonexcedance set of A122111.
%Y A352515 A352523 counts comps by unfixed pts, first col A010054, rank stat A352513.
%Y A352515 Cf. A088218, A098825, A114088, A131577, A238352.
%K A352515 nonn
%O A352515 0,4
%A A352515 _Gus Wiseman_, Mar 23 2022