A242428
Length of longest chain of nonempty proper subsemigroups of the dual symmetric inverse monoid.
Original entry on oeis.org
0, 2, 17, 180, 3298, 88431, 3064050, 130905678, 6732227475, 409094032964, 28917250469178, 2346562701385648, 216180120430479731, 22397392442055209003, 2588479398843886168171, 331352273262513644199134, 46692196905193286953380160, 7203294536351261350956567853, 1210694223244114528129261255186
Offset: 1
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a[n_] := Sum[StirlingS2[n, i] (i! (StirlingS2[n, i] - 1)/2 - DigitCount[i, 2, 1] + Ceiling[3 i/2] + 1), {i, 1, n}] - n - 1;
Array[a, 19] (* Jean-François Alcover, Dec 12 2018, from PARI *)
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b(n)=if(n<1, 0, b(n\2)+n%2) /* A000120 */
a(n)=-n-1+sum(i=1, n, stirling(n,i,flag=2)*(ceil(3*i/2)-b(i)+1+(stirling(n,i,flag=2)-1)*i!/2))
A242429
Length of longest chain of nonempty proper subsemigroups of the monoid of partial injective order-preserving functions of a chain with n elements.
Original entry on oeis.org
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
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a[n_] := Binomial[2n, n]/2 + 3*2^(n-1) - n - 2; Array[a, 30] (* Jean-François Alcover, Dec 15 2018, from PARI *)
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a(n)=-2-n+sum(i=0, n, binomial(n,i)*(binomial(n,i)+3)/2);
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
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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 *)
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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))
Showing 1-3 of 3 results.