A194460 a(n) is the number of basic ideals in the standard Borel subalgebra of the untwisted affine Lie algebra sl_n.

1, 4, 18, 82, 370, 1648, 7252, 31582, 136338, 584248, 2488156, 10540484, 44450068, 186715072, 781628008, 3262239862, 13579324498, 56391614632, 233686316428, 966556003132, 3990942300508, 16453094542432, 67733512006168

Offset: 1

Author

Volodymyr Mazorchuk, Aug 24 2011

Keywords

• nonn

Comments

a(n) also equals the number of pairs (p,q) of Dyck paths of semilength n, such that the first peak of q has height at least n-l(p), where l(p) is the height of the last peak of p, and the last peak of q has height at least n-f(p), where f(p) is the height of the first peak of p.

From Per Alexandersson, May 26 2018: (Start)

a(n) is also equal to the number of circular arc digraphs on n vertices.

a(n) is equal to the number of lists b(1),b(2),...,b(n) such that 0 <= b(i) < n and b(i)-1 <= b(i+1) for i=1..n-1 and b(n)-1 <= b(1).

The subset of such sequences such that b(n)=0 is given by the Catalan numbers, A000108. (End)

Christian Krattenthaler has shown that a(n) = (n+2)*binomial(2*n-1,n-1) - 2^(2*n-1), which also implies the above recursion observed by D. S. McNeil. - Volodymyr Mazorchuk, Aug 26 2011

Examples

			G.f. = x + 4*x^2 + 18*x^3 + 82*x^4 + 370*x^5 + 1648*x^6 + 7252*x^7 + 31582*x^8 + ... - _Michael Somos_, Jun 28 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Per Alexandersson, Svante Linusson, Samu Potka, The cyclic sieving phenomenon on circular Dyck paths, arXiv:1903.01327 [math.CO], 2019.
• Per Alexandersson and Greta Panova, LLT polynomials, chromatic quasisymmetric functions and graphs with cycles, arXiv:1705.10353 [math.CO], 2017. See Lemma 5.
• K. Baur and V. Mazorchuk; Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras, arXiv:1108.3659 [math.RA], 2011.

Programs

Magma

[(n+2)*Binomial(2*n-1, n-1) - 2^(2*n-1): n in [1..30]]; // G. C. Greubel, Aug 13 2018

Mathematica

a[n_] := (n+2) Binomial[2n-1, n-1] - 2^(2n-1);
Array[a, 23] (* Jean-François Alcover, Jul 27 2018, after Michael Somos *)

PARI

{a(n) = if( n<1, 0, (n+2) * binomial(2*n-1, n-1) - 2^(2*n-1))}; /* Michael Somos, Jun 28 2018 */

Sage

def A194460(n):
    if n == 1: return 1
    cf = CachedFunction(lambda i,j,n: binomial(n-1-i+n-1-j,n-i-1)-binomial(n-1-i+n-1-j, n-i-j-1))
    CP = cartesian_product
    return sum(sum(cf(i,j,n)*cf(k,m,n) for k,m in CP([[n-i..n],[n-j..n]])) for i,j in CP([[1..n],[1..n]]))
# D. S. McNeil, Aug 25 2011

Formula

It appears that the sequence is given by a(1)=1, a(n) = 4*a(n-1) + 2*binomial(2*n-3, n-3). - D. S. McNeil, Aug 25 2011

0 = a(n)*(+2304*a(n+1) -3744*a(n+2) +1464*a(n+3) -168*a(n+4)) +a(n+1)*(-96*a(n+1) +1192*a(n+2) -730*a(n+3) +102*a(n+4)) +a(n+2)*(-78*a(n+2) +99*a(n+3) -19*a(n+4)) +a(n+3)*(-3*a(n+3) +a(n+4)) for all n>0. - Michael Somos, Jun 28 2018

Extensions

More terms from D. S. McNeil, Aug 25 2011