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

%I A051288 #62 Mar 17 2025 09:37:01
%S A051288 1,2,4,2,8,12,16,48,6,32,160,60,64,480,360,20,128,1344,1680,280,256,
%T A051288 3584,6720,2240,70,512,9216,24192,13440,1260,1024,23040,80640,67200,
%U A051288 12600,252,2048,56320,253440,295680,92400,5544,4096,135168,760320
%N A051288 Triangle read by rows: T(n,k) = number of paths of n upsteps U and n downsteps D that contain k UUDs.
%C A051288 By reading paths backward, the UUD in the name could be replaced by DDU.
%C A051288 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.
%C A051288 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
%C A051288 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_
%C A051288 From _Colin Defant_, Sep 16 2018: (Start)
%C A051288 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.
%C A051288 T(n,k) is the number of permutations of [n] with k valleys that avoid the patterns 1342, 3142, 3412, and 3421. (End)
%C A051288 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
%H A051288 Stanislav Sykora, <a href="/A051288/b051288.txt">Table of n, a(n) for n = 0..2600</a>
%H A051288 Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%H A051288 Rui Duarte and António Guedes de Oliveira, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Duarte/duarte7.html">A Famous Identity of Hajós in Terms of Sets</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.
%H A051288 S. Elizalde, E. Deutsch, <a href="https://doi.org/10.54550/ECA2022V2S1R7">The degree of asymmetry of a sequence</a>, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7 eq (10)
%H A051288 Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, <a href="https://doi.org/10.37236/2181">Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property</a>, Electr. J. Combinatorics, 19 (2012), #P2.
%H A051288 Stanislav Sykora, <a href="/A051288/a051288.txt">Triangle T(n,k) for rows n = 0..100</a>
%H A051288 Stanislav Sykora, <a href="http://www.ebyte.it/library/educards/math/CombiTransitions.html">p-Quantum Transitions and a Combinatorial Identity</a>, Stan's Library II, 2007, Identity (1) for p=0.
%H A051288 Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
%H A051288 Hans J. H. Tuenter, <a href="https://doi.org/10.1080/0025570X.2024.2402197">Combinatorial Analysis of a Classic Card Trick</a>, Mathematics Magazine, 97(5):551-558, December 2024.
%F A051288 T(n, k) = binomial(n, 2*k)*2^(n-2*k)*binomial(2*k, k).
%F A051288 G.f.: (1-4*x+4*x^2*(1-y))^(-1/2) = Sum_{n>=0, k>=0} a(n, k)*x^n*y^k.
%e A051288 Table begins
%e A051288 n | k=0    1    2    3
%e A051288 --+-------------------
%e A051288 0 |   1
%e A051288 1 |   2
%e A051288 2 |   4    2
%e A051288 3 |   8   12
%e A051288 4 |  16   48    6
%e A051288 5 |  32  160   60
%e A051288 6 |  64  480  360   20
%e A051288 7 | 128 1344 1680  280
%e A051288 ...
%e A051288 a(2,1)=2 because UUDD, DUUD each have one UUD.
%t A051288 Table[Binomial[n, 2k]2^(n-2k)Binomial[2k, k], {n, 0, 15}, {k, 0, n/2}]
%Y A051288 Row sums are the (even) central binomial coefficients A000984. A091894 gives the distribution of the parameter "number of DDUs" on Dyck paths.
%K A051288 nonn,tabf
%O A051288 0,2
%A A051288 _Clark Kimberling_
%E A051288 Additional comments from _David Callan_, Aug 28 2004