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A381299 Irregular triangular array read by rows. T(n,k) is the number of ordered set partitions of [n] with exactly k descents, n>=0, 0<=k<=binomial(n,2).

%I A381299 #57 Aug 23 2025 16:42:12
%S A381299 1,1,2,1,4,4,4,1,8,12,18,18,12,6,1,16,32,60,84,100,92,76,48,24,8,1,32,
%T A381299 80,176,300,448,572,650,658,596,478,334,206,102,40,10,1,64,192,480,
%U A381299 944,1632,2476,3428,4300,5008,5372,5356,4936,4220,3316,2392,1556,904,456,188,60,12,1
%N A381299 Irregular triangular array read by rows.  T(n,k) is the number of ordered set partitions of [n] with exactly k descents, n>=0, 0<=k<=binomial(n,2).
%C A381299 Let p = ({b_1},{b_2},...,{b_m}) be an ordered set partition of [n] into m blocks for some m, 1<=m<=n. A descent in p is an ordered pair (x,y) in [n]X[n] such that x is in b_i, y is in b_j, i<j, and x>y.
%C A381299 T(n,binomial(n,2)) = 1 (counts the ordered set partition ({n},{n-1},...,{2},{1})).
%C A381299 For n>=1, T(n,0) = 2^(n-1).
%C A381299 Sum_{k>=0} T(n,k)*2^k = A289545(n).
%C A381299 Sum_{k>=0} T(n,k)*3^k = A347841(n).
%C A381299 Sum_{k>=0} T(n,k)*4^k = A347842(n).
%C A381299 Sum_{k>=0} T(n,k)*5^k = A347843(n).
%C A381299 Sum_{k>=0} T(n,k)*6^k = A385408(n).
%C A381299 Sum_{k>=0} T(n,k)*7^k = A347844(n).
%C A381299 Sum_{k>=0} T(n,k)*8^k = A347845(n).
%C A381299 Sum_{k>=0} T(n,k)*9^k = A347846(n).
%C A381299 T(n,k) is the number of preferential arrangements of n labeled elements with exactly k inversions. For example, there 4 preferential rearrangements of length 3 with 1 inversion: 132, 213, 212, 131. - _Kyle Celano_, Aug 18 2025
%H A381299 Alois P. Heinz, <a href="/A381299/b381299.txt">Rows n = 0..50, flattened</a>
%H A381299 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://arxiv.org/abs/1909.01550">Counting acyclic and strong digraphs by descents</a>, arXiv:1909.01550 [math.CO], 2019-2020.
%H A381299 Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, <a href="https://arxiv.org/abs/2508.11587">Inversions in parking functions</a>, arXiv:2508.11587 [math.CO], 2025. See Theorem 1.
%F A381299 Sum_{k=0..binomial(n,2)} k * T(n,k) = A240796(n). - _Alois P. Heinz_, Feb 20 2025
%F A381299 T(n,k) is the coefficient of q^k in n!_q times the coefficient of x^n in 1/(1- x - x^2/2!_q - x^3/3!_q - ...), where n!_q= 1*(1+q)*(1+q+q^2)*...*(1+q+...+q^(n-1)). - _Ira M. Gessel_, Jun 24 2025
%F A381299 T(n,k) = Sum_{w} 2^(asc(w)), where w runs through the set of permutations with k inversions and asc(w) is the number of ascents of w. - _Kyle Celano_, Aug 18 2025
%e A381299 Triangle T(n,k) begins:
%e A381299   1;
%e A381299   1;
%e A381299   2,  1;
%e A381299   4,  4,  4,  1;
%e A381299   8, 12, 18, 18,  12,  6,  1;
%e A381299  16, 32, 60, 84, 100, 92, 76, 48, 24, 8, 1;
%e A381299  ...
%p A381299 b:= proc(o, u, t) option remember; expand(`if`(u+o=0, 1, `if`(t=1,
%p A381299       b(u+o, 0$2), 0)+add(x^(u+j-1)*b(o-j, u+j-1, 1), j=1..o)))
%p A381299     end:
%p A381299 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
%p A381299 seq(T(n), n=0..8);  # _Alois P. Heinz_, Feb 21 2025
%t A381299 nn = 7; B[n_] := FunctionExpand[QFactorial[n, u]]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[1/(1 -(e[z] - 1)), {z, 0, nn}], z]] // Grid
%Y A381299 Columns k=0-2 give: A011782, A001787(n-1) for n>=1, 2*A268586.
%Y A381299 Cf. A000670 (row sums), A008302 (the cases where each block has size 1).
%Y A381299 Cf. A125810, A161680, A240796, A289545, A347841, A347842, A347843, A347844, A347845, A347846, A385408.
%K A381299 nonn,tabf,changed
%O A381299 0,3
%A A381299 _Geoffrey Critzer_, Feb 19 2025