A352489 -id:A352489 - cp's OEIS frontend

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

Gus Wiseman, Mar 23 2022

nonn

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.

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

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

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.

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}]

A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256, 264, 270, 280

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than or equal to that of their conjugate.

			The terms together with their prime indices begin:
    1: ()
    2: (1)
    4: (1,1)
    6: (2,1)
    8: (1,1,1)
    9: (2,2)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   20: (3,1,1)
   24: (2,1,1,1)
   27: (2,2,2)
   30: (3,2,1)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   56: (4,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A046682.
The opposite version is A352489, strong A352487.
The strong version is A352490, counted by A000701.
These are the positions of nonnegative terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352525 counts compositions by weak superdiagonals, rank statistic A352517.
Cf. A000720, A045931, A114088, A120383, A175508, A290822, A319005, A325040, A325698, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#>=Times@@Prime/@conj[primeMS[#]]&]

a(n) >= A122111(a(n)).

cp's OEIS Frontend

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

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