A352517 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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.