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

%I A352487 #18 Mar 25 2022 21:23:34
%S A352487 3,5,7,10,11,13,14,15,17,19,21,22,23,25,26,28,29,31,33,34,35,37,38,39,
%T A352487 41,42,43,44,45,46,47,49,51,52,53,55,57,58,59,61,62,63,65,66,67,68,69,
%U A352487 70,71,73,74,76,77,78,79,82,83,85,86,87,88,89,91,92,93,94
%N A352487 Excedance set of A122111. Numbers k < A122111(k), where A122111 represents partition conjugation using Heinz numbers.
%C A352487 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.
%H A352487 Alois P. Heinz, <a href="/A352487/b352487.txt">Table of n, a(n) for n = 1..10000</a>
%H A352487 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 A352487 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%F A352487 a(n) < A122111(a(n)).
%e A352487 The terms together with their prime indices begin:
%e A352487    3: (2)
%e A352487    5: (3)
%e A352487    7: (4)
%e A352487   10: (3,1)
%e A352487   11: (5)
%e A352487   13: (6)
%e A352487   14: (4,1)
%e A352487   15: (3,2)
%e A352487   17: (7)
%e A352487   19: (8)
%e A352487   21: (4,2)
%e A352487   22: (5,1)
%e A352487   23: (9)
%e A352487   25: (3,3)
%e A352487   26: (6,1)
%e A352487   28: (4,1,1)
%e A352487 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.
%t A352487 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352487 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352487 Select[Range[100],#<Times@@Prime/@conj[primeMS[#]]&]
%Y A352487 These partitions are counted by A000701.
%Y A352487 The weak version is A352489, counted by A046682.
%Y A352487 The opposite version is A352490, weak A352488.
%Y A352487 These are the positions of negative terms in A352491.
%Y A352487 A000041 counts integer partitions, strict A000009.
%Y A352487 A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
%Y A352487 A003963 = product of prime indices, conjugate A329382.
%Y A352487 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352487 A008480 counts permutations of prime indices, conjugate A321648.
%Y A352487 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352487 A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
%Y A352487 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352487 A173018 counts permutations by excedances, weak A123125.
%Y A352487 A238744 = partition conjugate of prime signature, ranked by A238745.
%Y A352487 A330644 counts non-self-conjugate partitions, ranked by A352486.
%Y A352487 A352521 counts compositions by subdiagonals, rank statistic A352514.
%Y A352487 Cf. A000720, A114088, A120383, A175508, A290822, A304360, A316524, A319005, A325040.
%K A352487 nonn
%O A352487 1,1
%A A352487 _Gus Wiseman_, Mar 19 2022