|
%I #50 Mar 26 2024 16:16:01
%S 1,1,1,2,2,1,4,6,3,1,9,16,12,4,1,21,45,40,20,5,1,51,126,135,80,30,6,1,
%T 127,357,441,315,140,42,7,1,323,1016,1428,1176,630,224,56,8,1,835,
%U 2907,4572,4284,2646,1134,336,72,9,1,2188,8350,14535,15240,10710,5292,1890,480,90,10,1
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks at even height.
%C Number of ordered trees with n edges having k leaves at even height. Row sums are the Catalan numbers (A000108). T(n,0)=A001006(n-1) (the Motzkin numbers). Sum_{k=0..n-1} k*T(n,k) = binomial(2n-2, n-2) = A001791(n-1). Mirror image of A091187.
%C T(n,k) is the number of Dyck paths of semilength n and having k dud's (here u=(1,1) and d=(1,-1)). Example: T(4,2)=3 because we have uud(du[d)ud], uu(dud)(dud) and uu(du[d)ud]d (the dud's are shown between parentheses).
%C T(n,k) is the number of Dyck paths of semilength n and containing exactly k double rises whose matching down steps form a doublefall. Example: UUUDUDDD has 2 double rises but only the first has matching Ds - the path's last 2 steps - forming a doublefall. (Travel horizontally east from an up step to encounter its matching down step.) - _David Callan_, Jul 15 2004
%C T(n,k) is the number of ordered trees on n edges containing k edges of outdegree 1. (The outdegree of an edge is the outdegree of its child vertex. Thus edges of outdegree 1 correspond to non-root vertices of outdegree 1.) T(3,2)=2 because
%C /\.../\.
%C |.....|.
%C each have one edge of outdegree 1. - _David Callan_, Oct 25 2004
%C Exponential Riordan array [exp(x)*Bessel_I(1,2x)/x, x]. - _Paul Barry_, Mar 09 2010
%C T(n, k) is the number of Dyck paths of semilength n and having k udu's (here u=(1,1) and d=(1,-1)). Note that reversing a path swaps u and d, thus udu becomes dud and vice versa. - _Michael Somos_, Feb 26 2020
%H Alois P. Heinz, <a href="/A091869/b091869.txt">Rows n = 1..200, flattened</a>
%H J. L. Baril and S. Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, Preprint, 2016. See Table 2.
%H David Callan, <a href="https://arxiv.org/abs/1702.06150">Bijections for Dyck paths with all peak heights of the same parity</a>, arXiv:1702.06150 [math.CO], 2017.
%H M. Dziemianczuk, <a href="http://dx.doi.org/10.1016/j.disc.2014.07.024">Enumerations of plane trees with multiple edges and Raney lattice paths</a>, Discrete Mathematics 337 (2014): 9-24.
%H A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%H Yidong Sun, <a href="http://dx.doi.org/10.1016/j.disc.2004.07.002">The statistic "number of udu's" in Dyck paths</a>, Discrete Math., 287 (2004), 177-186.
%H Chao-Jen Wang, <a href="http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf">Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords</a>.
%F T(n, k) = binomial(n-1, k)*(Sum_{j=0..ceiling((n-k)/2)} binomial(n-k, j)*binomial(n-k-j, j-1))/(n-k) for 0 <= k < n; T(n, k)=0 for k >= n.
%F G.f.: G = G(t, z) satisfies z*G^2 - (1 + z - t*z)*G + 1 + z - t*z = 0. T(n, k) = M(n-k-1)*binomial(n-1, k), where M(n) = A001006(n) are the Motzkin numbers.
%F T(n+1, k+1) = n*T(n, k)/(k+1). - _David Callan_, Dec 09 2004
%F G.f.: 1/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-... (continued fraction). - _Paul Barry_, Aug 03 2009
%F E.g.f.: exp(x+xy)*Bessel_I(1,2x)/x. - _Paul Barry_, Mar 10 2010
%e T(4,1)=6 because we have u(ud)dudud, udu(ud)dud, ududu(ud)d, uuudd(ud)d, u(ud)uuddd and uuu(ud)ddd (here u=(1,1), d=(1,-1) and the peaks at even height are shown between parentheses).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 4, 6, 3, 1;
%e 9, 16, 12, 4, 1;
%e 21, 45, 40, 20, 5, 1;
%e 51, 126, 135, 80, 30, 6, 1;
%e ...
%p T := proc(n,k) if k<n then binomial(n-1,k)*sum(binomial(n-k,j)*binomial(n-k-j,j-1),j=0..ceil((n-k)/2))/(n-k) else 0 fi end: seq(seq(T(n,k),k=0..n-1),n=1..11);
%p # second Maple program:
%p b:= proc(x, y, t) option remember; expand(`if`(x=0, 1,
%p `if`(y>0, b(x-1, y-1, 0)*z^irem(t*y+t, 2), 0)+
%p `if`(y<x-1, b(x-1, y+1, 1), 0)))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
%p seq(T(n), n=1..16); # _Alois P. Heinz_, May 12 2017
%t (* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, n_] = 1; t[n_, k_] := m[n - k]*Binomial[n - 1, k - 1]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 10 2013 *)
%o (PARI) {T(n, k) = my(y, c, w); if( k<0 || k>=n, 0, w = vector(n); forvec(v=vector(2*n, k, [0, 1]), c=y=0; for(k=1, 2*n, if( 0>(y += (-1)^v[k]), break)); if( y, next); for(i=1, 2*n-2, c += ([0, 1, 0] == v[i..i+2])); w[c+1]++); w[k+1])}; /* _Michael Somos_, Feb 26 2020 */
%Y Cf. A000108, A001006, A001791, A091187, A243752.
%Y T(2n,n) gives A371411.
%K nonn,tabl
%O 1,4
%A _Emeric Deutsch_, Mar 10 2004
|
