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

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, 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, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1

Offset

0,21

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 5392th composition in standard order is (2,2,4,5), with excedances {1,3,4}, so a(5392) = 3.

Mathematica

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

Crossrefs

Positions of first appearances are A104462.

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

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

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

The triangle A352524 counts these compositions (first column A008930).

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.

A352487 is the 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: A325674 A055641 A218245 * A086075 A316865 A325616

Adjacent sequences: A352513 A352514 A352515 * A352517 A352518 A352519

Keyword

nonn

Author

Gus Wiseman, Mar 23 2022

Status

approved