cp's OEIS Frontend

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

%I A352489 #12 Mar 25 2022 21:27:34
%S A352489 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,
%T A352489 34,35,37,38,39,41,42,43,44,45,46,47,49,51,52,53,55,56,57,58,59,61,62,
%U A352489 63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85
%N A352489 Weak excedance set of A122111. Numbers k <= A122111(k), where A122111 represents partition conjugation using Heinz numbers.
%C A352489 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.
%H A352489 Alois P. Heinz, <a href="/A352489/b352489.txt">Table of n, a(n) for n = 1..10000</a>
%H A352489 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%H A352489 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.
%F A352489 a(n) <= A122111(a(n)).
%e A352489 The terms together with their prime indices begin:
%e A352489    1: ()
%e A352489    2: (1)
%e A352489    3: (2)
%e A352489    5: (3)
%e A352489    6: (2,1)
%e A352489    7: (4)
%e A352489    9: (2,2)
%e A352489   10: (3,1)
%e A352489   11: (5)
%e A352489   13: (6)
%e A352489   14: (4,1)
%e A352489   15: (3,2)
%e A352489   17: (7)
%e A352489   19: (8)
%e A352489   20: (3,1,1)
%e A352489 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.
%t A352489 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352489 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352489 Select[Range[100],#<=Times@@Prime/@conj[primeMS[#]]&]
%Y A352489 These partitions are counted by A046682.
%Y A352489 The strong version is A352487, counted by A000701.
%Y A352489 The opposite version is A352488, strong A352490
%Y A352489 These are the positions of nonpositive terms in A352491.
%Y A352489 A000041 counts integer partitions, strict A000009.
%Y A352489 A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
%Y A352489 A003963 = product of prime indices, conjugate A329382.
%Y A352489 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352489 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352489 A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
%Y A352489 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352489 A173018 counts permutations by excedances, weak A123125.
%Y A352489 A330644 counts non-self-conjugate partitions, ranked by A352486.
%Y A352489 A352522 counts compositions by weak subdiagonals, rank statistic A352515.
%Y A352489 Cf. A000720, A096276, A114088, A120383, A301987, A321648, A324850, A325040, A325044.
%K A352489 nonn
%O A352489 1,2
%A A352489 _Gus Wiseman_, Mar 20 2022