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

%I A352490 #11 Mar 25 2022 21:25:36
%S A352490 4,8,12,16,18,24,27,32,36,40,48,50,54,60,64,72,80,81,90,96,100,108,
%T A352490 112,120,128,135,140,144,150,160,162,168,180,192,196,200,216,224,225,
%U A352490 240,243,250,252,256,270,280,288,300,315,320,324,336,352,360,375,378
%N A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.
%C A352490 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.
%H A352490 Alois P. Heinz, <a href="/A352490/b352490.txt">Table of n, a(n) for n = 1..2000</a>
%H A352490 Richard Ehrenborg and Einar Steingrímsson, <a href="https://www.ms.uky.edu/~jrge/Papers/Excedance.pdf">The Excedance Set of a Permutation</a>, Advances in Applied Mathematics 24, (2000), 284-299.
%H A352490 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%F A352490 a(n) > A122111(a(n)).
%e A352490 The terms together with their prime indices begin:
%e A352490     4: (1,1)
%e A352490     8: (1,1,1)
%e A352490    12: (2,1,1)
%e A352490    16: (1,1,1,1)
%e A352490    18: (2,2,1)
%e A352490    24: (2,1,1,1)
%e A352490    27: (2,2,2)
%e A352490    32: (1,1,1,1,1)
%e A352490    36: (2,2,1,1)
%e A352490    40: (3,1,1,1)
%e A352490    48: (2,1,1,1,1)
%e A352490    50: (3,3,1)
%e A352490    54: (2,2,2,1)
%e A352490    60: (3,2,1,1)
%e A352490    64: (1,1,1,1,1,1)
%e A352490 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.
%t A352490 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352490 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352490 Select[Range[100],#>Times@@Prime/@conj[primeMS[#]]&]
%Y A352490 These partitions are counted by A000701.
%Y A352490 The opposite version is A352487, weak A352489.
%Y A352490 The weak version is A352488, counted by A046682.
%Y A352490 These are the positions of positive terms in A352491.
%Y A352490 A000041 counts integer partitions, strict A000009.
%Y A352490 A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
%Y A352490 A003963 = product of prime indices, conjugate A329382.
%Y A352490 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352490 A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
%Y A352490 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352490 A173018 counts permutations by excedances, weak A123125.
%Y A352490 A330644 counts non-self-conjugate partitions, ranked by A352486.
%Y A352490 A352521 counts compositions by subdiagonals, rank statistic A352514.
%Y A352490 Cf. A000720, A114088, A114324, A321648, A324850, A325037, A325038, A325040.
%K A352490 nonn
%O A352490 1,1
%A A352490 _Gus Wiseman_, Mar 20 2022