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

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, 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, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 2, 3, 3

Offset: 0

Author

Gus Wiseman, Mar 23 2022

Keywords

• nonn

Comments

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.

Examples

			The 89th composition in standard order is (2,1,3,1), with weak nonexcedances {2,3,4}, so a(89) = 3.
		

Links

• MathOverflow, Why 'excedances' of permutations? [closed].

Crossrefs

Positions of first appearances are A000225.

The strong version is A352514, counted by A352521 (first column A219282).

The strong opposite version is A352516, counted by A352524 (first A008930).

The opposite version is A352517, counted by A352525 (first column A177510).

Triangle A352522 counts these comps (first col A238874), partitions A115994.

A008292 is the triangle of Eulerian numbers (version without zeros).

A011782 counts compositions.

A173018 counts permutations by number of excedances, weak A123125.

A238349 counts comps by fixed points, first col A238351, rank stat A352512.

A352488 is the weak nonexcedance set of A122111.

A352523 counts comps by unfixed pts, first col A010054, rank stat A352513.

Cf. A088218, A098825, A114088, A131577, A238352.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
paw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[paw[stc[n]],{n,0,30}]