A352515 - OEIS

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

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OFFSET

0,4

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

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}]

CROSSREFS

Positions of first appearances are A000225.

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