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 A247285 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. 1
 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, 1, 13, 85, 300, 543, 408, 79, 1, 1, 15, 117, 523, 1335, 1771, 966, 133, 1, 1, 17, 154, 833, 2747, 5303, 5335, 2184, 221, 1, 1, 19, 196, 1244, 5031, 12792, 19272, 15099, 4767, 364, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. Number of entries in row n is n. Sum of entries in row n is the Catalan number A000108(n). Sum(k*T(n,k), k>=0) = A172061(n-2). 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). T(n+1,n) = A001924(n) for n>=1. - Alois P. Heinz, Sep 11 2014 LINKS Alois P. Heinz, Rows n = 1..141, flattened A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176. Y. Le Borgne, Counting upper interactions in Dyck paths, Sem. Lotharingien de Combinatoire, 54, 2006, Article B54f. FORMULA 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 It is extremely complex; the Maple program follows it (blindly), except that the infinite sums have been replaced by summations from n=0 to n=15. EXAMPLE 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. Triangle starts: 1; 1,1; 1,3,1; 1,5,7,1; 1,7,19,14,1; 1,9,36,59,26,1; MAPLE 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 # second Maple program: b:= proc(x, y, t, c) option remember; `if`(y<0 or 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))))) end: T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(2*n, 0, false, 0)): seq(T(n), n=1..15); # Alois P. Heinz, Sep 11 2014 MATHEMATICA 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 *) CROSSREFS Cf. A000108, A001924, A172061, A091866. Sequence in context: A145661 A119258 A099608 * A047969 A047812 A129392 Adjacent sequences: A247282 A247283 A247284 * A247286 A247287 A247288 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Sep 11 2014 STATUS approved

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Last modified September 12 07:57 EDT 2024. Contains 375850 sequences. (Running on oeis4.)