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A381529 T(n,k) is the number of permutations of [n] having exactly k pairs of integers i in [n] such that their cycle minima have opposite sorting order; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows.

%I A381529 #34 Feb 27 2025 17:05:26
%S A381529 1,1,2,5,1,15,5,4,54,21,24,16,5,235,89,118,112,101,35,28,2,1237,408,
%T A381529 577,633,719,585,402,239,167,59,14,7790,2106,3023,3529,4410,4463,4600,
%U A381529 3012,2789,1933,1438,629,442,122,34,57581,12529,17693,20980,27208,30064,35359,33332,28137,24970,22850,17148,14272,8645,5639,3684,1809,664,282,34
%N A381529 T(n,k) is the number of permutations of [n] having exactly k pairs of integers i<j in [n] such that their cycle minima have opposite sorting order; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows.
%H A381529 Alois P. Heinz, <a href="/A381529/b381529.txt">Rows n = 0..55, flattened</a>
%H A381529 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%H A381529 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A381529 Sum_{k>=1} k * T(n,k) = A126673(n)/2.
%e A381529 T(4,0) = 15: (1)(2)(3)(4), (1,2)(3)(4), (1)(2,3)(4), (1)(2)(3,4), (1,2)(3,4), (1,2,3)(4), (1,3,2)(4), (1)(2,3,4), (1)(2,4,3), (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2).
%e A381529 T(4,1) = 5: (1)(2,4)(3), (1,2,4)(3), (1,4,2)(3), (1,3)(2)(4), (1,3)(2,4).
%e A381529 T(4,2) = 4: (1,4)(2)(3), (1,4)(2,3), (1,3,4)(2), (1,4,3)(2).
%e A381529 Triangle T(n,k) begins:
%e A381529      1;
%e A381529      1;
%e A381529      2;
%e A381529      5,   1;
%e A381529     15,   5,   4;
%e A381529     54,  21,  24,  16,   5;
%e A381529    235,  89, 118, 112, 101,  35,  28,   2;
%e A381529   1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14;
%e A381529   ...
%p A381529 b:= proc(o, u, t) option remember; expand(`if`(u+o=0, max(0, t-1)!,
%p A381529      `if`(t>0, b(u+o, 0$2)*(t-1)!, 0)+add(x^(u+j-1)*
%p A381529         b(o-j, u+j-1, t+1), j=`if`(t=0, 1, 1..o))))
%p A381529     end:
%p A381529 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
%p A381529 seq(T(n), n=0..10);
%Y A381529 Columns k=0-1 give: A051295, A381539.
%Y A381529 Row sums give A000142.
%Y A381529 Row lengths give A125811.
%Y A381529 Last elements of rows give A381531.
%Y A381529 Main diagonal gives A381545.
%Y A381529 Cf. A008302, A125810 (similar for set partitions), A126673, A381299 (similar for ordered set partitions).
%K A381529 nonn,tabf
%O A381529 0,3
%A A381529 _Alois P. Heinz_, Feb 26 2025