login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A129175 Triangle read by rows: MacMahon's q-analog of the Catalan numbers. 7

%I #49 Mar 07 2023 16:38:46

%S 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,

%T 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,

%U 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

%N Triangle read by rows: MacMahon's q-analog of the Catalan numbers.

%C 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.

%C 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'.

%C Row n has 1+n*(n-1) terms.

%C Row sums are the Catalan numbers (A000108).

%C T(n,k) = T(n,n^2-n-k) (i.e., rows are palindromic).

%C Alternating row sums are the central binomial coefficients binomial(n, floor(n/2)) = A001405(n).

%C Sum_{k=0..n*(n-1)} k*T(n,k) = A002740(n+1).

%C T(n,k) = A129174(n,n+k) (i.e., except for the initial 0's, rows of A129174 and A129175 are the same).

%C 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

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

%D P. A. MacMahon, Combinatory analysis, Two volumes (bound as one), Chelsea Publishing Co., New York, 1960 (see p. 214).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 b. [From _Emeric Deutsch_, Nov 05 2008]

%H Alois P. Heinz, <a href="/A129175/b129175.txt">Rows n = 0..32, flattened</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000027">The major index of a Dyck path.</a>

%H J. Furlinger and J. Hofbauer, <a href="http://dx.doi.org/10.1016/0097-3165(85)90089-5">q-Catalan numbers</a>, J. Comb. Theory, A, 40, 248-264, 1985.

%H M. Shattuck, <a href="http://www.emis.de/journals/INTEGERS/papers/f7/f7.Abstract.html">Parity theorems for statistics on permutations and Catalan words</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.

%F 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).

%e 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).

%e Triangle starts:

%e 1;

%e 1;

%e 1,0,1;

%e 1,0,1,1,1,0,1;

%e 1,0,1,1,2,1,2,1,2,1,1,0,1;

%e 1,0,1,1,2,2,3,2,4,3,4,3,4,2,3,2,2,1,1,0,1;

%p 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

%p # second Maple program:

%p b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,

%p expand(b(x-1, y-1, true)+b(x-1, y+1, false)*`if`(t, z^x, 1))))

%p end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, false)):

%p seq(T(n), n=0..8); # _Alois P. Heinz_, Sep 15 2014

%t 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_ *)

%t 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 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_catalan_number

%o def T(n): return list(q_catalan_number(n))

%o for n in (0..6): print(T(n)) # _Peter Luschny_, Jul 19 2016

%Y Cf. A000108, A001405, A002740, A129174, A227543.

%K nonn,tabf

%O 0,17

%A _Emeric Deutsch_, Apr 20 2007

%E New name from _Peter Luschny_, Jul 24 2016

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)