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A367594 Number of permutations of [n] whose cycle maxima sum to k, where k is chosen so as to maximize this number.

%I A367594 #49 Oct 03 2024 07:54:25
%S A367594 1,1,1,2,7,27,142,834,5962,46788,426708,4198632,46516800,551415936,
%T A367594 7197404976,99712618560,1500173940960,23786129681280,405087689727360,
%U A367594 7237524061198080,137652562628778240,2735042530132523520,57482464477451489280,1257272784581092070400
%N A367594 Number of permutations of [n] whose cycle maxima sum to k, where k is chosen so as to maximize this number.
%H A367594 Alois P. Heinz, <a href="/A367594/b367594.txt">Table of n, a(n) for n = 0..400</a>
%H A367594 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A367594 a(n) = A143947(n,2n-1) for n>=1, a(0) = 1.
%e A367594 a(4) = 7 = A143947(4,7): (123)(4), (132)(4), (124)(3), (142)(3), (13)(24),
%e A367594   (14)(23), (1)(2)(34).
%e A367594 a(5) = 27 = A143947(5,9): (1234)(5), (1243)(5), (1324)(5), (1342)(5), (1423)(5), (1432)(5), (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (124)(35), (142)(35), (125)(34), (152)(34), (134)(25), (143)(25), (135)(24), (153)(24), (14)(235), (14)(253), (15)(234), (15)(243), (1)(23)(45), (1)(245)(3), (1)(254)(3).
%p A367594 b:= proc(n) option remember;
%p A367594       `if`(n=0, 1, expand(b(n-1)*(t-n+x^n)))
%p A367594     end:
%p A367594 a:= n-> max(coeffs(subs(t=n, b(n)))):
%p A367594 seq(a(n), n=0..23);
%t A367594 a[n_] := If[n == 0, 1, Module[{t}, CoefficientList[Product[n-k+t^k, {k, 1, n-1}]*t^(n-1), t] // Max]];
%t A367594 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 03 2024 *)
%Y A367594 Row maxima of A143947.
%Y A367594 Cf. A368678.
%K A367594 nonn
%O A367594 0,4
%A A367594 _Alois P. Heinz_, Jan 03 2024