A202058 Number of ascent sequences avoiding the pattern 000.

1, 1, 2, 4, 10, 27, 83, 277, 1015, 4007, 17047, 77451, 374889, 1923168, 10427250, 59544957, 357236992, 2245822801, 14762969601, 101264286082, 723499803180, 5375063821727, 41459660565329, 331546282841906, 2745163969235517, 23505333233440927, 207895424692608432

Offset: 0

Author

N. J. A. Sloane, Dec 10 2011

Keywords

• nonn

Comments

It appears that no formula or g.f. is known.

Links

• Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..176
• Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
• P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011

Crossrefs

Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317; 0000 is A317784.

Column k=2 of A294220.

Programs

Mathematica

b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n==0, 1, Sum[If[Coefficient[ p, x, j]==k, 0, b[n-1, j, t + If[j>i, 1, 0], p+x^j, k]], {j, 1, t+1}]];
a[n_] := b[n, 0, 0, 0, Min[n, 2]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Sep 01 2018, after Alois P. Heinz in A294220 *)

Extensions

a(15)-a(17) from Alois P. Heinz, Nov 09 2012

a(18)-a(20) from Giovanni Resta, Jan 06 2014

a(21) from Vaclav Kotesovec, Aug 21 2018

a(22) from Vaclav Kotesovec, Aug 22 2018

More terms from Anthony Guttmann, Nov 04 2021