cp's OEIS Frontend

A140717 Triangle read by rows: T(n,k) is the number of Dyck paths d of semilength n such that sum of peakheights of d - number of peaks of d = k (n >= 0, 0 <= k <= floor(n^2/4)).

%I A140717 #26 Sep 03 2024 01:41:21
%S A140717 1,1,1,1,1,2,2,1,3,5,4,1,1,4,9,12,10,4,2,1,5,14,25,31,26,16,9,4,1,1,6,
%T A140717 20,44,70,82,74,54,38,22,12,4,2,1,7,27,70,134,196,227,215,179,139,99,
%U A140717 64,38,20,9,4,1,1,8,35,104,231,400,558,644,641,576,488,384,288,200,134,80
%N A140717 Triangle read by rows: T(n,k) is the number of Dyck paths d of semilength n such that sum of peakheights of d - number of peaks of d = k (n >= 0, 0 <= k <= floor(n^2/4)).
%C A140717 T(n,k) is the number of 321-avoiding permutations of {1,2,...,n} having inversion number equal to k. Example: T(4,2) = 5 because we have 1423, 1342, 3124, 2143 and 2341.
%C A140717 Conjecture: antidiagonal sums equal A227309. - _Mikhail Kurkov_, Aug 30 2024
%H A140717 Alois P. Heinz, <a href="/A140717/b140717.txt">Rows n = 0..50, flattened</a>
%H A140717 E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, <a href="https://doi.org/10.1080/10236199908808200">ECO: a methodology for the enumeration of combinatorial objects</a>, Journal of Difference Equations and Applications, 5, 1999, 435-490.
%H A140717 E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, <a href="https://doi.org/10.1016/S0012-365X(00)00359-9">Some permutations with forbidden subsequences and their inversion number</a>, Discrete Math., 234, 2001, 1-15.
%H A140717 Emeric Deutsch, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00371-9">Dyck path enumeration</a>, Discrete Math., 204, 1999, 167-202 (see section 5).
%H A140717 G. Feinberg and K.-H. Lee, <a href="http://arxiv.org/abs/1401.0845">Homogeneous representations of KLR-algebras and fully commutative elements</a>, arXiv preprint arXiv:1401.0845 [math.RT], 2014-2015.
%H A140717 Niket Gowravaram and Tanya Khovanova, <a href="http://arxiv.org/abs/1509.00462">On the Structure of nil-Temperley-Lieb Algebras of type A</a>, arXiv:1509.00462 [math.CO], 2015.
%F A140717 G.f.: G(t,z) = H(t,1/t,z), where H(t,x,z) = 1 + zH(t,x,z)[H(t,tx,z)-1+tx] (H(t,x,z) is the trivariate g.f. of Dyck paths with respect to semilength, sum of peak-heights and number of peaks, marked by z, t and x, respectively).
%F A140717 Sum_{k>=0} k*T(n,k) = A008549(n-1).
%F A140717 Row n has 1 + floor(n^2/4) entries.
%F A140717 Conjecture: n-th row polynomial equals t_n for n > 0 where we start with vector v of fixed length m with elements v_i = 1, then set t = v and for i=1..m-1, for j=i+1..m apply [v_i, v_j] := [v_i + z^(j-i)*v_j, z*v_i + v_j] (here square brackets mean that instead of sequentially assigning v_i and then v_j, we reserve their values (for example, as A = v_i, B = v_j) and then assign them in any order) and t_{i+1} := v_{i+1} (after ending each cycle for j). It also looks like that if we change z^(j-i) to z^(2*(j-i)) it gives us equivalence of t_n and n-th row polynomial of A227543. - _Mikhail Kurkov_, Aug 30 2024
%e A140717 T(4,2) = 5 because we have UDUUDUDD (5 - 3 = 2), UDUUUDD (4 - 2 = 2), UUDDUUDD (4 - 2 = 2), UUDUDDUD (5 - 3 = 2) and UUUDDDUD (4 - 2 = 2); here U = (1,1), D = (1,-1).
%e A140717 Triangle starts:
%e A140717   1;
%e A140717   1;
%e A140717   1, 1;
%e A140717   1, 2,  2;
%e A140717   1, 3,  5,  4,  1;
%e A140717   1, 4,  9, 12, 10,  4,  2;
%e A140717   1, 5, 14, 25, 31, 26, 16, 9, 4, 1;
%p A140717 H := 1/(1+z-t*x*z-z*h[1]):
%p A140717 for n to 13 do h[n]:=1/(1+z-x*t^(n+1)*z-z*h[n+1]) end do:
%p A140717 G := subs({h[11]=0,x=1/t},H): Gser := simplify(series(G,z=0,12)):
%p A140717 for n from 0 to 9 do P[n] := sort(coeff(Gser,z,n)) end do:
%p A140717 for n from 0 to 9 do seq(coeff(P[n],t,j), j=0..floor((1/4)*n^2)) end do;
%p A140717 # yields sequence in triangular form
%t A140717 m = rows = 10; mt = 2 m + 1; mx = mz = m - 1;
%t A140717 H[_, _, _] = 0; Do[H[t_, x_, z_] = Series[1 + z (H[t, t x, z] - 1 + t x) H[t, x, z], {t, 0, mt}, {x, 0, mx}, {z, 0, mz}] // Normal, {m}];
%t A140717 G[t_, z_] = Series[H[t, 1/t, z], {t, 0, mt}, {z, 0, mz}] // Normal // Collect[#, z]&;
%t A140717 CoefficientList[#, t]& /@ CoefficientList[G[t, z], z] // Take[#, m]& // Flatten (* _Jean-François Alcover_, Nov 25 2018 *)
%Y A140717 Row sums are the Catalan numbers A000108.
%Y A140717 Cf. A008549, A129183, A227309, A227543.
%K A140717 nonn,tabf
%O A140717 0,6
%A A140717 _Emeric Deutsch_, Jun 08 2008