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A343488 Irregular table T(n, k), n >= 0, k = 1..max(1, n), read by rows; T(n, k) is the number of permutations s of { 1..n } such that p(s) = k where p(s) is the least m > 0 such that, working in Z/nZ, s(i) + m = s(i + m) for i = 1..n.

%I A343488 #8 Apr 18 2021 09:04:00
%S A343488 1,1,2,0,3,0,3,4,4,0,16,5,0,0,0,115,6,12,42,0,0,660,7,0,0,0,0,0,5033,
%T A343488 8,24,0,352,0,0,0,39936,9,0,153,0,0,0,0,0,362718,10,40,0,0,3830,0,0,0,
%U A343488 0,3624920,11,0,0,0,0,0,0,0,0,0,39916789,12,60,372,1872,0,45636,0,0,0,0,0,478953648
%N A343488 Irregular table T(n, k), n >= 0, k = 1..max(1, n), read by rows; T(n, k) is the number of permutations s of { 1..n } such that p(s) = k where p(s) is the least m > 0 such that, working in Z/nZ, s(i) + m = s(i + m) for i = 1..n.
%C A343488 We set the row for n = 0 to [1] by convention.
%C A343488 The number p(s) can be interpreted as the period of the permutation s.
%H A343488 Igor Pak, <a href="https://www.math.ucla.edu/~pak/papers/inf2.pdf">Periodic permutations and the Robinson-Schensted correspondence</a>
%H A343488 Rémy Sigrist, <a href="/A343488/a343488.txt">C program for A343488</a>
%H A343488 <a href="https://oeis.org/index/Per#perm">Index entries for sequences related to permutations</a>
%F A343488 T(n, 1) = max(n, 1).
%F A343488 T(n, n) = A324514(n).
%F A343488 Sum_{k = 1..max(1, n)} T(n, k) = n!.
%e A343488 Table begins:
%e A343488      0:     [1]
%e A343488      1:     [1]
%e A343488      2:     [2, 0]
%e A343488      3:     [3, 0, 3]
%e A343488      4:     [4, 4, 0, 16]
%e A343488      5:     [5, 0, 0, 0, 115]
%e A343488      6:     [6, 12, 42, 0, 0, 660]
%e A343488      7:     [7, 0, 0, 0, 0, 0, 5033]
%e A343488      8:     [8, 24, 0, 352, 0, 0, 0, 39936]
%e A343488      9:     [9, 0, 153, 0, 0, 0, 0, 0, 362718]
%e A343488     10:     [10, 40, 0, 0, 3830, 0, 0, 0, 0, 3624920]
%o A343488 (C) See Links section.
%Y A343488 Cf. A000142, A324514.
%K A343488 nonn,tabf
%O A343488 0,3
%A A343488 _Rémy Sigrist_, Apr 17 2021