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A276067 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having length of first descent k (n>=2, 1<=k<=n-1). A descent is a maximal sequence of consecutive down steps. 1

%I

%S 1,1,1,2,2,1,5,4,3,1,14,9,7,4,1,41,23,16,11,5,1,122,64,39,27,16,6,1,

%T 366,186,103,66,43,22,7,1,1105,552,289,169,109,65,29,8,1,3356,1657,

%U 841,458,278,174,94,37,9,1,10251,5013,2498,1299,736,452,268,131,46,10,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having length of first descent k (n>=2, 1<=k<=n-1). A descent is a maximal sequence of consecutive down steps.

%C Number of entries in row n is n-1.

%C Sum of entries in row n = A082582(n).

%C Sum(k*T(n,k),k>=0) = A276068(n).

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.

%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016

%F G.f.: G(t,z) = t(1-2z)(1-2z-z^2-Q)/(z(1-z-tz)), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

%F T(n,k)= T(n-1,k)+T(n-1,k-1) (n>=3, k>=2).

%e Row 4 is 2,2,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that the lengths of their first descents are 1,2,1,2,3, respectively.

%e Triangle starts

%e 1;

%e 1,1;

%e 2,2,1;

%e 5,4,3,1;

%e 14,9,7,4,1.

%p G := (1/2)*t*(1-2*z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-z-t*z)): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 17 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 17 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form

%t m = maxExponent = 13;

%t G = ((1/2) t (1 - 2z)(1 - 2z - z^2 - Sqrt[(1 - z)(1 - 3z - z^2 - z^3)])/ (z(1 - z - t z)) + O[z]^m) + O[t]^m;

%t Drop[CoefficientList[#/t, t]& /@ CoefficientList[G, z], 2] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)

%Y Cf. A082582, A276068.

%K nonn,tabl

%O 2,4

%A _Emeric Deutsch_, Aug 25 2016

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Last modified March 26 05:43 EDT 2019. Contains 321481 sequences. (Running on oeis4.)