A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

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

Offset

0,7

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.

Links

Table of n, a(n) for n=0..86.

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

Example

The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.

Mathematica

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

Crossrefs

Positive positions of first appearances are A164894.

The version for partitions is A257990.

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

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

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

The triangle A352525 counts these compositions (first column A177510).

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.

A352489 is the weak excedance set of A122111.

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

Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

Sequence in context: A230850 A072085 A054868 * A347981 A065081 A366643

Adjacent sequences: A352514 A352515 A352516 * A352518 A352519 A352520

Keyword

nonn

Author

Gus Wiseman, Mar 23 2022

Status

approved