

A051288


Triangle read by rows: T(n,k) = number of paths of n upsteps U and n downsteps D that contain k UUDs.


11



1, 2, 4, 2, 8, 12, 16, 48, 6, 32, 160, 60, 64, 480, 360, 20, 128, 1344, 1680, 280, 256, 3584, 6720, 2240, 70, 512, 9216, 24192, 13440, 1260, 1024, 23040, 80640, 67200, 12600, 252, 2048, 56320, 253440, 295680, 92400, 5544, 4096, 135168, 760320
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OFFSET

0,2


COMMENTS

By reading paths backward, the UUD in the name could be replaced by DDU.
Or, triangular array T read by rows: T(n,k)=P(2n,n,4k), where P(n,k,c)=number of vectors (x(1),x(2,),...,x(n)) of k 1's and nk 0's such that x(i)=x(n+1i) for exactly c values of i. P(n,k,n) counts palindromes.
In nuclear magnetic resonance of n coupled spin1/2 nuclides, T(n,k) is the number of zeroquantum transitions with combination index k. See the [Sykora (2007)] link, containing also yet another interpretation in terms of pairs of binary ntuples. [Stanislav Sykora Apr 27 2012]
Let u  (u_1, u_2, u_3, ..., u_{2n}) be a binary vector containing n 0's and n 1's. Define a mismatch to be an adjacent pair (u_{2i1}, u_{2i}) which is neither 0,1 nor 1,0 (think "socks"). Then T(n,k) = number of u's with k mismatches.  N. J. A. Sloane, Nov 03 2017 following an email from Bill Gosper.
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stacksorting map. T(n,k) is the number of permutations pi of [n] with k valleys such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n] with k valleys such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n] with k valleys that avoid the patterns 1342, 3142, 3412, and 3421. (End)


LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2600
Colin Defant, Stacksorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.
Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, Kostas Manes, Strings of Length 3 in GrandDyck Paths and the ChungFeller Property, Electr. J. Combinatorics, 19 (2012), #P2.
Stanislav Sykora, Triangle T(n,k) for rows n = 0..100
Stanislav Sykora, pQuantum Transitions and a Combinatorial Identity, Stan's Library II, 2007, Identity (1) for p=0


FORMULA

T(n, k) = binom(n, 2*k)*2^(n2*k)*binom(2*k, k).
G.f.: (14*x+4*x^2*(1y))^(1/2) = Sum_{n>=0, k>=0} a(n, k)*x^n*y^k.


EXAMPLE

Table begins
n  k=0 1 2 3
+
0  1
1  2
2  4 2
3  8 12
4  16 48 6
5  32 160 60
6  64 480 360 20
7  128 1344 1680 280
...
a(2,1)=2 because UUDD, DUUD each have one UUD.


MATHEMATICA

Table[Binomial[n, 2k]2^(n2k)Binomial[2k, k], {n, 0, 15}, {k, 0, n/2}]


CROSSREFS

Row sums are the (even) central binomial coefficients A000984. A091894 gives the distribution of the parameter "number of DDUs" on Dyck paths.
Sequence in context: A114593 A114655 A228890 * A120434 A319030 A285335
Adjacent sequences: A051285 A051286 A051287 * A051289 A051290 A051291


KEYWORD

nonn,tabf


AUTHOR

Clark Kimberling


EXTENSIONS

Additional comments from David Callan, Aug 28 2004


STATUS

approved



