A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

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

Offset: 0

Author

Gus Wiseman, Mar 22 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 83rd composition in standard order is (2,3,1,1), with strong nonexcedances {3,4}, so a(83) = 2.
		

Links

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

Crossrefs

Positions of first appearances are A000225.

The weak version is A352515, counted by A352522 (first column A238874).

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

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

The triangle A352521 counts these compositions (first column A219282).

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 parts, first col A238351, rank stat A352512.

A352490 is the (strong) nonexcedance set of A122111.

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

Cf. A088218, A098825, A114088, A115994, A131577, A238352.

Programs

Mathematica

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