A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 50, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 315, 320, 324, 336, 352, 360, 375, 378

Offset

1,1

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 greater than that of their conjugate.

Links

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].

Formula

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

Example

The terms together with their prime indices begin:
    4: (1,1)
    8: (1,1,1)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   24: (2,1,1,1)
   27: (2,2,2)
   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)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
For example, the partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, and 196 > 189, so 196 is in the sequence, and 189 is not.

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[#]]&]

Crossrefs

These partitions are counted by A000701.

The opposite version is A352487, weak A352489.

The weak version is A352488, counted by A046682.

These are the positions of positive 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.

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.

A352521 counts compositions by subdiagonals, rank statistic A352514.

Cf. A000720, A114088, A114324, A321648, A324850, A325037, A325038, A325040.

Sequence in context: A272405 A311119 A321177 * A340788 A113645 A311120

Adjacent sequences: A352487 A352488 A352489 * A352491 A352492 A352493

Keyword

nonn

Author

Gus Wiseman, Mar 20 2022

Status

approved