A242428 -id:A242428 - cp's OEIS frontend

A242429 Length of longest chain of nonempty proper subsemigroups of the monoid of partial injective order-preserving functions of a chain with n elements.

1, 5, 17, 53, 167, 550, 1899, 6809, 25067, 93902, 355775, 1358208, 5212573, 20082860, 77607895, 300638481, 1166999699, 4537960846, 17673418311, 68924837252, 269132082925, 1052055773292, 4116727946687, 16123827007348, 63205353550497, 247959367137320, 973469914150619, 3824345703033374, 15033634055076857

Offset: 1

James Mitchell, May 14 2014

nonn

James Mitchell, Table of n, a(n) for n = 1..100
P. J. Cameron, M. Gadouleau, J. D. Mitchell, Y. Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015.

Cf. A227914, A242428.

a[n_] := Binomial[2n, n]/2 + 3*2^(n-1) - n - 2; Array[a, 30] (* Jean-François Alcover, Dec 15 2018, from PARI *)

a(n)=-2-n+sum(i=0, n, binomial(n,i)*(binomial(n,i)+3)/2);

Conjecture: n*(131*n-376)*a(n) +2*(-563*n^2+1993*n-1185)*a(n-1) +3*(1099*n^2-4678*n+4684)*a(n-2) +2*(-1987*n^2+9803*n-12021)*a(n-3) +4*(209*n-387)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Oct 20 2015

a(n) = binomial(2*n,n)/2 + 3*2^(n-1) - n - 2. - Gheorghe Coserea, May 16 2016

A242432 Length of longest chain of nonempty proper subsemigroups of the monoid of partial injective orientation-preserving functions of a chain with n elements.

Original entry on oeis.org

1, 6, 24, 92, 363, 1483, 6191, 26077, 109987, 462900, 1941613, 8115138, 33805905, 140413073, 581694265, 2404314784, 9917782935, 40837958578, 167889571658, 689231516287, 2825851058202, 11572537702747, 47342211484912, 193485587828057, 790066214186999, 3223470297388819, 13141840760544209, 53540833421980514

Offset: 1

James Mitchell, May 14 2014

nonn

James Mitchell, Table of n, a(n) for n = 1..100
P. J. Cameron, M. Gadouleau, J. D. Mitchell, Y. Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015.

Cf. A001222, A227914, A242428, A242429.

b[n_] := If[n < 1, 0, PrimeOmega[n]];
a[n_] := -2 - n + Sum[Binomial[n, i]*(b[i] + (Binomial[n, i] - 1)*i/2 + 2), {i, 0, n}];
Array[a, 28] (* Jean-François Alcover, Feb 19 2019, from PARI *)

b(n)=if(n<1, 0, bigomega(n)) /* A001222 */
a(n)=-2-n+sum(i=0, n, binomial(n,i)*(b(i)+(binomial(n,i)-1)*i/2+2))

cp's OEIS Frontend

A242429 Length of longest chain of nonempty proper subsemigroups of the monoid of partial injective order-preserving functions of a chain with n elements.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula