A114503 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n for which height of first peak + height of last peak = k (n>=1; 2<=k<=2n).

1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 4, 4, 2, 1, 0, 1, 5, 10, 11, 8, 4, 2, 1, 0, 1, 14, 28, 32, 26, 16, 8, 4, 2, 1, 0, 1, 42, 84, 98, 84, 57, 32, 16, 8, 4, 2, 1, 0, 1, 132, 264, 312, 276, 198, 120, 64, 32, 16, 8, 4, 2, 1, 0, 1, 429, 858, 1023, 924, 687, 438, 247, 128, 64, 32, 16, 8, 4, 2, 1, 0, 1

Offset: 1

Author

Emeric Deutsch, Dec 02 2005

Keywords

• nonn
• tabf

Comments

Row n has 2n-1 terms. Column 2 yields the Catalan numbers (A000108). T(n,3)=2T(n,2) (n>=3). Sum(kT(n,k),k=2..2n)=2[Catalan(n+1)-Catalan(n)] (A071721). The trivariate g.f., with z marking semilength, t marking height of the first peak and s marking height of the last peak, is G = (1-tzC-szC+tsz^2*C^2+tsz^2*C)/[(1-tzC)(1-szC)(1-tsz)]-1.

Examples

			T(5,6)=4 because we have UUDUUUDDDD, UUUDUDUDDD, UUUDDUUDDD and UUUUDDDUDD, where U=(1,1), D=(1,-1).
Triangle starts:
  1;
  1,0,1;
  1,2,1,0,1;
  2,4,4,2,1,0,1;
  5,10,11,8,4,2,1,0,1;
  ...
		

Links

• Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.

Crossrefs

Cf. A000108, A071721.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: g:=(1-2*t*z*C+t^2*z^2*C^2+t^2*z^2*C)/(1-t*z*C)^2/(1-t^2*z)-1: gser:=simplify(series(g,z=0,12)): for n from 1 to 10 do P[n]:=coeff(gser,z^n) od: for n from 1 to 10 do seq(coeff(P[n],t^j),j=2..2*n) od; # yields sequence in triangular form

Formula

G.f.: (1-2tzC+t^2*z^2*C^2+t^2*z^2*C)/[(1-tzC)^2*(1-t^2*z)]-1, where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.