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

%I A352488 #12 Mar 25 2022 21:20:36
%S A352488 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,
%T A352488 81,84,90,96,100,108,112,120,125,128,135,140,144,150,160,162,168,176,
%U A352488 180,192,196,200,210,216,224,225,240,243,250,252,256,264,270,280
%N A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.
%C A352488 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.
%H A352488 Alois P. Heinz, <a href="/A352488/b352488.txt">Table of n, a(n) for n = 1..2000</a>
%H A352488 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 A352488 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%F A352488 a(n) >= A122111(a(n)).
%e A352488 The terms together with their prime indices begin:
%e A352488     1: ()
%e A352488     2: (1)
%e A352488     4: (1,1)
%e A352488     6: (2,1)
%e A352488     8: (1,1,1)
%e A352488     9: (2,2)
%e A352488    12: (2,1,1)
%e A352488    16: (1,1,1,1)
%e A352488    18: (2,2,1)
%e A352488    20: (3,1,1)
%e A352488    24: (2,1,1,1)
%e A352488    27: (2,2,2)
%e A352488    30: (3,2,1)
%e A352488    32: (1,1,1,1,1)
%e A352488    36: (2,2,1,1)
%e A352488    40: (3,1,1,1)
%e A352488    48: (2,1,1,1,1)
%e A352488    50: (3,3,1)
%e A352488    54: (2,2,2,1)
%e A352488    56: (4,1,1,1)
%t A352488 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352488 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352488 Select[Range[100],#>=Times@@Prime/@conj[primeMS[#]]&]
%Y A352488 These partitions are counted by A046682.
%Y A352488 The opposite version is A352489, strong A352487.
%Y A352488 The strong version is A352490, counted by A000701.
%Y A352488 These are the positions of nonnegative terms in A352491.
%Y A352488 A000041 counts integer partitions, strict A000009.
%Y A352488 A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
%Y A352488 A003963 = product of prime indices, conjugate A329382.
%Y A352488 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352488 A008480 counts permutations of prime indices, conjugate A321648.
%Y A352488 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352488 A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
%Y A352488 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352488 A173018 counts permutations by excedances, weak A123125.
%Y A352488 A330644 counts non-self-conjugate partitions, ranked by A352486.
%Y A352488 A352525 counts compositions by weak superdiagonals, rank statistic A352517.
%Y A352488 Cf. A000720, A045931, A114088, A120383, A175508, A290822, A319005, A325040, A325698, A347450.
%K A352488 nonn
%O A352488 1,2
%A A352488 _Gus Wiseman_, Mar 20 2022