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A336070 Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).

%I A336070 #42 Jan 18 2025 09:08:35
%S A336070 1,1,2,6,23,106,567,3440,23286,173704,1414102,12465119,118205428,
%T A336070 1199306902,12958274048,148502304614,1798680392716,22953847041950,
%U A336070 307774885768354,4325220458515307,63563589415836532,974883257009308933,15575374626562632462,258780875395778033769,4464364292401926006220
%N A336070 Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).
%C A336070 From _Joerg Arndt_, Jan 20 2024: (Start)
%C A336070 a(n) is the number of weak ascent sequences of length n.
%C A336070 A weak ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) counts the weak ascents d(j) >= d(j-1) of its argument.
%C A336070 The number of length-n weak ascent sequences with maximal number of weak ascents is A000108(n).
%C A336070 (End)
%H A336070 Alois P. Heinz, <a href="/A336070/b336070.txt">Table of n, a(n) for n = 0..400</a>
%H A336070 Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/2003.11533">Wilf equivalences between vincular patterns in inversion sequences</a>, arXiv:2003.11533 [math.CO], 2020. See p. 5, Table 1. Gives terms 1-10.
%H A336070 Beata Benyi, Anders Claesson, and Mark Dukes, <a href="https://arxiv.org/abs/2111.03159">Weak ascent sequences and related combinatorial structures</a>, arXiv:2111.03159 [math.CO], (4-November-2021).
%e A336070 From _Joerg Arndt_, Jan 20 2024: (Start)
%e A336070 There are a(4) = 23 weak ascent sequences (dots for zeros):
%e A336070    1:  [ . . . . ]
%e A336070    2:  [ . . . 1 ]
%e A336070    3:  [ . . . 2 ]
%e A336070    4:  [ . . . 3 ]
%e A336070    5:  [ . . 1 . ]
%e A336070    6:  [ . . 1 1 ]
%e A336070    7:  [ . . 1 2 ]
%e A336070    8:  [ . . 1 3 ]
%e A336070    9:  [ . . 2 . ]
%e A336070   10:  [ . . 2 1 ]
%e A336070   11:  [ . . 2 2 ]
%e A336070   12:  [ . . 2 3 ]
%e A336070   13:  [ . 1 . . ]
%e A336070   14:  [ . 1 . 1 ]
%e A336070   15:  [ . 1 . 2 ]
%e A336070   16:  [ . 1 1 . ]
%e A336070   17:  [ . 1 1 1 ]
%e A336070   18:  [ . 1 1 2 ]
%e A336070   19:  [ . 1 1 3 ]
%e A336070   20:  [ . 1 2 . ]
%e A336070   21:  [ . 1 2 1 ]
%e A336070   22:  [ . 1 2 2 ]
%e A336070   23:  [ . 1 2 3 ]
%e A336070 (End)
%p A336070 b:= proc(n, i, t) option remember; `if`(n=0, 1,
%p A336070       add(b(n-1, j, t+`if`(j>=i, 1, 0)), j=0..t+1))
%p A336070     end:
%p A336070 a:= n-> b(n, -1$2):
%p A336070 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 23 2024
%t A336070 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[b[n - 1, j, t + If[j >= i, 1, 0]], {j, 0, t + 1}]];
%t A336070 a[n_] := b[n, -1, -1];
%t A336070 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 18 2025, after _Alois P. Heinz_ *)
%o A336070 (PARI) \\ see formula (5) on page 18 of the Benyi/Claesson/Dukes reference
%o A336070 N=40;
%o A336070 M=matrix(N,N,r,c,-1);  \\ memoization
%o A336070 a(n,k)=
%o A336070 {
%o A336070     if ( n==0 && k==0, return(1) );
%o A336070     if ( k==0, return(0) );
%o A336070     if ( n==0, return(0) );
%o A336070     if ( M[n,k] != -1 , return( M[n,k] ) );
%o A336070     my( s );
%o A336070     s = sum( i=0, n, sum( j=0, k-1,
%o A336070          (-1)^j * binomial(k-j,i) * binomial(i,j) * a( n-i, k-j-1 )) );
%o A336070     M[n,k] = s;
%o A336070     return( s );
%o A336070 }
%o A336070 for (n=0, N, print1( sum(k=1,n,a(n,k)),", "); );
%o A336070 \\ print triangle a(n,k), see A369321:
%o A336070 \\ for (n=0, N, for(k=0,n, print1(a(n,k),", "); ); print(););
%o A336070 \\ _Joerg Arndt_, Jan 20 2024
%Y A336070 Cf. A000079, A000108, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336071, A336072.
%Y A336070 Row sums of A369321.
%K A336070 nonn
%O A336070 0,3
%A A336070 _Michael De Vlieger_, Jul 07 2020
%E A336070 a(0)=1 prepended and more terms from _Joerg Arndt_, Jan 20 2024