A352489 Weak excedance set of A122111. Numbers k <= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Author

Gus Wiseman, Mar 20 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 or equal to that of their conjugate.

Examples

			The terms together with their prime indices begin:
   1: ()
   2: (1)
   3: (2)
   5: (3)
   6: (2,1)
   7: (4)
   9: (2,2)
  10: (3,1)
  11: (5)
  13: (6)
  14: (4,1)
  15: (3,2)
  17: (7)
  19: (8)
  20: (3,1,1)
For example, the partition (3,2,2) has Heinz number 45 and its conjugate (3,3,1) has Heinz number 50, and 45 <= 50, so 45 is in the sequence, and 50 is not.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• MathOverflow, Why 'excedances' of permutations? [closed].
• Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.

Crossrefs

These partitions are counted by A046682.

The strong version is A352487, counted by A000701.

The opposite version is A352488, strong A352490

These are the positions of nonpositive 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).

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.

A352522 counts compositions by weak subdiagonals, rank statistic A352515.

Cf. A000720, A096276, A114088, A120383, A301987, A321648, A324850, A325040, A325044.

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],#<=Times@@Prime/@conj[primeMS[#]]&]

Formula

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