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A364967 Number T(n,k) of permutations of [n] for which the difference between the longest and the shortest cycle length is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

%I A364967 #34 Dec 08 2023 07:11:48
%S A364967 1,1,2,3,3,10,6,8,25,45,20,30,176,60,250,90,144,721,861,770,1344,504,
%T A364967 840,6406,1778,7980,6300,8736,3360,5760,42561,23283,38808,75348,45360,
%U A364967 66240,25920,45360,436402,84150,363680,456120,708048,378000,572400,226800,403200
%N A364967 Number T(n,k) of permutations of [n] for which the difference between the longest and the shortest cycle length is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
%C A364967 T(0,0) = 1 by convention.
%H A364967 Alois P. Heinz, <a href="/A364967/b364967.txt">Rows n = 0..150, flattened</a>
%H A364967 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A364967 T(n,k) == 0 (mod k!).
%F A364967 Sum_{k=0..max(0,n-2)} T(n,k)/k! = A365229(n).
%e A364967 T(4,0) = 10: (1)(2)(3)(4), (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), (1432).
%e A364967 T(4,1) = 6: (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4), (14)(2)(3).
%e A364967 T(4,2) = 8: (1)(234), (1)(243), (123)(4), (132)(4), (124)(3), (142)(3), (134)(2), (143)(2).
%e A364967 Triangle T(n,k) begins:
%e A364967       1;
%e A364967       1;
%e A364967       2;
%e A364967       3,     3;
%e A364967      10,     6,     8;
%e A364967      25,    45,    20,    30;
%e A364967     176,    60,   250,    90,   144;
%e A364967     721,   861,   770,  1344,   504,   840;
%e A364967    6406,  1778,  7980,  6300,  8736,  3360,  5760;
%e A364967   42561, 23283, 38808, 75348, 45360, 66240, 25920, 45360;
%e A364967   ...
%p A364967 b:= proc(n, l, m) option remember; `if`(n=0, x^(m-l), add(
%p A364967      b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1)*(j-1)!, j=1..n))
%p A364967     end:
%p A364967 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
%p A364967 seq(T(n), n=0..12);
%t A364967 b[n_, l_, m_] := b[n, l, m] = If[n == 0, x^(m - l), Sum[b[n - j, Min[l, j], Max[m, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
%t A364967 T[n_] := CoefficientList[b[n, n, 0], x];
%t A364967 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 08 2023, after _Alois P. Heinz_ *)
%Y A364967 Row sums give A000142.
%Y A364967 Column k=0 gives A005225 (for n>=1).
%Y A364967 T(n+1,n-1) gives A001048(n) (for n>=1).
%Y A364967 Cf. A126074, A145877, A364971, A365229.
%K A364967 nonn,tabf
%O A364967 0,3
%A A364967 _Alois P. Heinz_, Aug 14 2023