A051288 Triangle read by rows: T(n,k) = number of paths of n upsteps U and n downsteps D that contain k UUDs.

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabf

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 n-k 0's such that x(i)=x(n+1-i) for exactly c values of i. P(n,k,n) counts palindromes.

In nuclear magnetic resonance of n coupled spin-1/2 nuclides, T(n,k) is the number of zero-quantum transitions with combination index k. See the [Sykora (2007)] link, containing also yet another interpretation in terms of pairs of binary n-tuples. - 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_{2i-1}, 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 stack-sorting 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)

T(n,k) is the number of card sequences for a balanced deck of 2n cards that end up with k pairs of cards in the black pile at the end of Stewart James' classic Miraskill card trick. See Table 2 in the [Tuenter (2024)] link. - Hans J. H. Tuenter, Dec 27 2024

Examples

			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.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2600
• Colin Defant, Stack-sorting 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.
• S. Elizalde, E. Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7 eq (10)
• Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2.
• Stanislav Sykora, Triangle T(n,k) for rows n = 0..100
• Stanislav Sykora, p-Quantum Transitions and a Combinatorial Identity, Stan's Library II, 2007, Identity (1) for p=0.
• Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
• Hans J. H. Tuenter, Combinatorial Analysis of a Classic Card Trick, Mathematics Magazine, 97(5):551-558, December 2024.

Crossrefs

Row sums are the (even) central binomial coefficients A000984. A091894 gives the distribution of the parameter "number of DDUs" on Dyck paths.

Programs

Mathematica

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

Formula

T(n, k) = binomial(n, 2*k)*2^(n-2*k)*binomial(2*k, k).

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

Extensions

Additional comments from David Callan, Aug 28 2004