Volodymyr Mazorchuk - cp's OEIS frontend

User: Volodymyr Mazorchuk

Volodymyr Mazorchuk has authored 2 sequences.

A193215 Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide.

1, 2, 3, 6, 14, 38, 113, 356, 1164, 3906, 13364, 46426, 163294, 580316, 2080475, 7515038, 27324014, 99920756, 367264130, 1356043388, 5027345564, 18706888196, 69841532210, 261545298848, 982175296016, 3697785571820, 13954630170720, 52776659865348, 200006396351216, 759386612309146, 2888310863702017

Offset: 1

Volodymyr Mazorchuk, Aug 26 2011

a(n+1) - a(n) = A000958(n) (this reduces to David Callan's comment on A000958(n) from Aug 23 2011).

The sequence gives the trace of the matrix describing the statistics of Dyck paths of semilength n with respect to the heights of the first and the last peaks, see the paper of Baur and Mazorchuk.

G. C. Greubel, Table of n, a(n) for n = 1..1000
K. Baur and V. Mazorchuk, Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras, arXiv:1108.3659 [math.RA], 2011.

for n from 1 by 1 to 100 do 1+sum(binomial(2*n-2-2*k, n-1-k)-binomial(2*n-2-2*k, n-1-2*k), k = 1 .. n-1) end do

Table[1+Sum[Binomial[2*n-2-2*k, n-1-k]-Binomial[2*n-2-2*k, n-1-2*k],{k,1,n-1}],{n,1,20}] (* Vaclav Kotesovec, Mar 21 2014 *)

a(n)=1+sum(k=1,n-1,binomial(2*n-2-2*k, n-1-k)-binomial(2*n-2-2*k, n-1-2*k));

a(n) = 1 + Sum_{i=1..n-1} A000958(i).
Recurrence: 2*n*(5*n-11)*a(n) = 3*(15*n^2 - 53*n + 40)*a(n-1) - 3*(5*n^2 - 21*n + 20)*a(n-2) - 2*(2*n-5)*(5*n-6)*a(n-3). - Vaclav Kotesovec, Mar 21 2014
a(n) ~ 5*4^n/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

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

Original entry on oeis.org

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

Volodymyr Mazorchuk, Aug 24 2011

nonn

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

			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

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.

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

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

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

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

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

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

cp's OEIS Frontend

User: Volodymyr Mazorchuk

A193215 Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

PARI

Formula