|
%I #24 May 27 2015 10:34:13
%S 1,1,1,1,3,1,1,5,7,1,1,7,19,14,1,1,9,36,59,26,1,1,11,58,150,162,46,1,
%T 1,13,85,300,543,408,79,1,1,15,117,523,1335,1771,966,133,1,1,17,154,
%U 833,2747,5303,5335,2184,221,1,1,19,196,1244,5031,12792,19272,15099,4767,364,1
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n (n>=1) having k (0<=k<=n-1) upper interactions.
%C An upper interaction in a Dyck path is an occurrence of a string d^k u^k for some k>=1; here u = (1,1) and d = (1,-1). For example, the Dyck path uu[d(du)u]dd has 2 upper interactions, shown between parentheses.
%C Number of entries in row n is n.
%C Sum of entries in row n is the Catalan number A000108(n).
%C Sum(k*T(n,k), k>=0) = A172061(n-2).
%C The statistic "number of lower interactions", mentioned in the Le Borgne reference is basically identical with the statistic "pyramid weight" of the Denise and Simion reference (see A091866 and the bottom of p. 8 of the Le Borgne reference).
%C T(n+1,n) = A001924(n) for n>=1. - _Alois P. Heinz_, Sep 11 2014
%H Alois P. Heinz, <a href="/A247285/b247285.txt">Rows n = 1..141, flattened</a>
%H A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.
%H Y. Le Borgne, <a href="http://www.emis.de/journals/SLC/wpapers/s54leborgne.html">Counting upper interactions in Dyck paths</a>, Sem. Lotharingien de Combinatoire, 54, 2006, Article B54f.
%F The g.f. A(t,u), where t marks semilength and u marks upper interactions, is given in Proposition 2 of the Le Borgne reference. It is extremely complex; the Maple program follows it (blindly), except that the infinite sums have been replaced by summations from n=0 to n=15.
%e Row 3 is 1,3,1. Indeed, the number of upper interactions in uuuddd, uududd, uuddud, uduudd, and ududud are 0, 1, 1, 1, and 2, respectively.
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,3,1;
%e 1,5,7,1;
%e 1,7,19,14,1;
%e 1,9,36,59,26,1;
%p q := u*t: s := ((1+t-2*q-sqrt((1-t)*(1-t-4*q+4*q^2)))*(1/2))/(t*(1-q)): Q := proc (x, n) options operator, arrow: product(1-q^k*x, k = 0 .. n-1) end proc: A := -t*add(((q-t)*s/(1-q))^n*q^(binomial(n+2, 2)-1)/(Q(q, n)*Q(q*t*s^2, n)), n = 0 .. 15)/add(((q-t)*s/(1-q))^n*q^binomial(n+2, 2)*(1-t*q^n*s)/(Q(q, n)*Q(q*t*s^2, n)*(1-q^n*s)*(1-q^(n+1)*s)), n = 0 .. 15): Aser := simplify(series(A, t = 0, 22)): for n to 16 do P[n] := sort(coeff(Aser, t, n)) end do: for n to 13 do seq(coeff(P[n], u, j), j = 0 .. n-1) end do; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(x, y, t, c) option remember; `if`(y<0 or y>x, 0,
%p `if`(x=0, 1, expand(b(x-1, y+1, false, max(0, c-1))*
%p `if`(c>0, z, 1)+b(x-1, y-1, true, 1+`if`(t, c, 0)))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(2*n, 0, false, 0)):
%p seq(T(n), n=1..15); # _Alois P. Heinz_, Sep 11 2014
%t b[x_, y_, t_, c_] := b [x, y, t, c] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, False, Max[0, c-1]]*If[c>0, z, 1] + b[x-1, y-1, True, 1 + If[t, c, 0] ] ] ] ]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, n-1}]][b[2*n, 0, False, 0]]; Table[T[n], {n, 1, 25}] // Flatten (* _Jean-François Alcover_, May 27 2015, after _Alois P. Heinz_ *)
%Y Cf. A000108, A001924, A172061, A091866.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Sep 11 2014
|
