OFFSET
0,2
COMMENTS
a(N,p) equals X_{N}(N+1,p) := T_{N,p} for alpha= 1 =beta and N>=p>=1 in the Derrida et al. 1992 reference. The one-point correlation functions <tau_{K}>_{N} for alpha= 1 =beta equal a(N,K)/C(N+1) with C(n)=A000108(n) (Catalan) in this reference. See also the Derrida et al. 1993 reference. In the Liggett 1999 reference mu_{N}{eta:eta(k)=1} of prop. 3.38, p. 275 is identical with <tau_{k}>_{N} and rho=0 and lambda=1.
Identity for each row n>=1: a(n,m)+a(n,n-m+1)= C(n+1), with C(n+1)=A000108(n+1)(Catalan) for every m=1..floor((n+1)/2). E.g., a(2k+1,k+1)=C(2*(k+1)).
REFERENCES
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (19) - (23), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eqs. (43), (44), pp. 1501-2 and eq.(81) with eqs.(80) and (81).
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, pp. 269, 275.
G. Schuetz and E. Domany, Phase Transitions in an Exactly Soluble one-Dimensional Exclusion Process, J. Stat. Phys. 72 (1993) 277-295, eq. (2.18), p. 283, with eqs. (2.13)-(2.15).
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 14.
Wolfdieter Lang, First 10 rows.
Hua Xin and Huan Xiong, Descents in the Grand Dyck paths and the Chung-Feller property, Australas. J. Combin. 94 (1) (2026), 177-194. See Table 1 at page 192.
FORMULA
a(n,m) = A028364(n,n-m), n>=m>=0, else 0.
G.f. for column m>=1 (without leading zeros): (c(x)^3)sum(C(m-1, k)*c(x)^k, k=0..m-1), with C(n, m) := (m+1)*binomial(2*n-m, n-m)/(n+1) (Catalan convolutions A033184); and for m=0: c^2(x), where c(x) is g.f. of A000108 (Catalan).
G.f. for diagonal sequences: see g.f. for columns of A028364.
EXAMPLE
Triangle begins:
1;
2, 1;
5, 3, 2;
14, 9, 7, 5;
42, 28, 23, 19, 14;
132, 90, 76, 66, 56, 42;
429, 297, 255, 227, 202, 174, 132;
1430, 1001, 869, 785, 715, 645, 561, 429;
4862, 3432, 3003, 2739, 2529, 2333, 2123, 1859, 1430;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, add(
expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
end:
T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10); # Alois P. Heinz, Nov 28 2015
MATHEMATICA
t[n_, k_] := Sum[ CatalanNumber[n - j]*CatalanNumber[j], {j, 0, k}]; Flatten[ Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}]] (* Jean-François Alcover, Jul 17 2013 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Feb 05 2002
STATUS
approved
