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A129175
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Triangle read by rows: MacMahon's q-analog of the Catalan numbers.
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7
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1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 6, 9, 7, 9, 8, 9, 7, 9, 6, 7, 5, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 13, 12, 16, 16, 19, 18, 22, 20, 23, 21, 23
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,17
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COMMENTS
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Previous name: T(n,k) is the number of Dyck words of length 2n having major index k (n >= 0, k >= 0). A Dyck word of length 2n is a word of n 0's and n 1's for which no initial segment contains more 1's than 0's. The major index of a Dyck word is the sum of the positions of those 1's that are followed by a 0.
Representing a Dyck word p of length 2n as a Dyck path p', the major index of p is equal to the sum of the abscissae of the valleys of p'.
Row n has 1+n*(n-1) terms.
Row sums are the Catalan numbers (A000108).
T(n,k) = T(n,n^2-n-k) (i.e., rows are palindromic).
Alternating row sums are the central binomial coefficients binomial(n, floor(n/2)) = A001405(n).
Sum_{k=0..n*(n-1)} k*T(n,k) = A002740(n+1).
For another q-analog of the Catalan numbers, due to Carlitz and Riordan, that enumerates Dyck paths by an area statistic see A227543. - Peter Bala, Feb 28 2023
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
P. A. MacMahon, Combinatory analysis, Two volumes (bound as one), Chelsea Publishing Co., New York, 1960 (see p. 214).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 b. [From Emeric Deutsch, Nov 05 2008]
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LINKS
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J. Furlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, A, 40, 248-264, 1985.
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FORMULA
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The generating polynomial for row n is P[n](t) = binomial[2n,n]/[n+1], where [n+1]=1+t+t^2+...+t^n and binomial[2n,n] is a Gaussian polynomial (due to MacMahon).
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EXAMPLE
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T(4,8)=2 because we have 01001101 (with 10's starting at positions 2 and 6) and 00101011 (with 10's starting at positions 3 and 5).
Triangle starts:
1;
1;
1,0,1;
1,0,1,1,1,0,1;
1,0,1,1,2,1,2,1,2,1,1,0,1;
1,0,1,1,2,2,3,2,4,3,4,3,4,2,3,2,2,1,1,0,1;
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MAPLE
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br:=n->sum(q^i, i=0..n-1): f:=n->product(br(j), j=1..n): cbr:=(n, k)->f(n)/f(k)/f(n-k): P:=n->sort(expand(simplify(cbr(2*n, n)/br(n+1)))): for n from 0 to 7 do seq(coeff(P(n), q, k), k=0..n*(n-1)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
expand(b(x-1, y-1, true)+b(x-1, y+1, false)*`if`(t, z^x, 1))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, false)):
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MATHEMATICA
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b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y-1, True] + b[x-1, y+1, False]*If[t, z^x, 1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, False]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)
p[n_] := QBinomial[2n, n, q]/QBinomial[n+1, 1, q]; Table[CoefficientList[p[n] // FunctionExpand, q], {n, 0, 9}] // Flatten (* Peter Luschny, Jul 22 2016 *)
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PROG
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(Sage)
from sage.combinat.q_analogues import q_catalan_number
def T(n): return list(q_catalan_number(n))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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