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A344791
a(n) is the number of 2-point antichains in the poset D_{2n+1} of type D, whose elements are compositions of 2n+1.
1
2, 10, 29, 66, 129, 228, 374, 580, 860, 1230, 1707, 2310, 3059, 3976, 5084, 6408, 7974, 9810, 11945, 14410, 17237, 20460, 24114, 28236, 32864, 38038, 43799, 50190, 57255, 65040, 73592, 82960, 93194, 104346, 116469, 129618, 143849, 159220, 175790, 193620
OFFSET
2,1
COMMENTS
A type D poset has elements which are compositions with four parts {x_{1}, x_{2}, x_{3}, x_{4}} satisfying:
x_{i} >= 0 for i = 1..4,
at least two of its elements are positive,
x_{2} = x_{4}, and x_{3} - x_{1} >= 0.
The partial order relation '<' is defined as follows: {x_{1}, x_{2}, x_{3}, x_{4}} < {x'_{1}, x'_{2}, x'_{3}, x'_{4}} if and only if x'_{1} <= x_{1}, x_{2} <= x'_{2}, x'_{3} <= x_{3}, and x_{4} <= x'_{4}.
The poset D_{2n+1} is a poset of type D where the compositions belonging to D_{2n+1} are compositions of the number 2n+1, with n >= 2. A nice property of this type of poset is that the Hasse diagram of the poset D_{2n+1} is equal to the Hasse diagram of the poset D_{2n}.
LINKS
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
FORMULA
a(n) = Sum_{i=1..n} Sum_{j=0..floor(i/2)} h(n,i,j)*(t(i)-2*t(j)), where h(n,i,j) = 0 if i=n and j=0; h(n,i,j) = n+1-i if i=2j and 1 <= j <= floor(n/2); h(n,i,j) = 1 otherwise. n >= 2 and t(i) = A000217(i) is the i-th triangular number.
G.f.: x^2*(x^2-2*x-2)/((x+1)*(x-1)^5). - Alois P. Heinz, May 29 2021
a(n) = (1 - (-1)^n - 8*n - 4*n^2 + 8*n^3 + 2*n^4)/32. - Stefano Spezia, Jun 02 2021
EXAMPLE
For instance, the first term of the sequence is for n=2: a(2)=2 means that there are two 2-point antichains in the poset D_{5}={{1,0,4,0},{2,0,3,0},{0,1,3,1},{0,2,1,2},{1,1,2,1}}, namely ({1,0,4,0}, {2,0,3,0}) and ({0,1,3,1}, {1,1,2,1}).
MATHEMATICA
Drop[CoefficientList[Series[x^2*(x^2 - 2 x - 2)/((x + 1) (x - 1)^5), {x, 0, 41}], x], 2] (* Michael De Vlieger, Mar 04 2022 *)
PROG
(PARI) a(n) = (n-1)*(n+1)*(n^2+4*n-1) >> 4; \\ Kevin Ryde, May 29 2021
CROSSREFS
Cf. A000217.
Sequence in context: A372871 A076438 A203551 * A192976 A264677 A101561
KEYWORD
nonn,easy
AUTHOR
Isaías David Marín Gaviria and Agustín Moreno Cañadas, May 28 2021
STATUS
approved