A352487 Excedance set of A122111. Numbers k < A122111(k), where A122111 represents partition conjugation using Heinz numbers.

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94

Offset: 1

Author

Gus Wiseman, Mar 19 2022

Keywords

• nonn

Comments

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 less than that of their conjugate.

Examples

			The terms together with their prime indices begin:
   3: (2)
   5: (3)
   7: (4)
  10: (3,1)
  11: (5)
  13: (6)
  14: (4,1)
  15: (3,2)
  17: (7)
  19: (8)
  21: (4,2)
  22: (5,1)
  23: (9)
  25: (3,3)
  26: (6,1)
  28: (4,1,1)
For example, the partition (4,1,1) has Heinz number 28 and its conjugate (3,1,1,1) has Heinz number 40, and 28 < 40, so 28 is in the sequence, and 40 is not.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• 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].

Crossrefs

These partitions are counted by A000701.

The weak version is A352489, counted by A046682.

The opposite version is A352490, weak A352488.

These are the positions of negative 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.

A238744 = partition conjugate of prime signature, ranked by A238745.

A330644 counts non-self-conjugate partitions, ranked by A352486.

A352521 counts compositions by subdiagonals, rank statistic A352514.

Cf. A000720, A114088, A120383, A175508, A290822, A304360, A316524, A319005, A325040.

Programs

Mathematica

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],#
				

Formula

a(n) < A122111(a(n)).