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A336070
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Number of inversion sequences avoiding the vincular pattern 10-0 (or 10-1).
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6
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1, 1, 2, 6, 23, 106, 567, 3440, 23286, 173704, 1414102, 12465119, 118205428, 1199306902, 12958274048, 148502304614, 1798680392716, 22953847041950, 307774885768354, 4325220458515307, 63563589415836532, 974883257009308933, 15575374626562632462, 258780875395778033769, 4464364292401926006220
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of weak ascent sequences of length n.
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.
The number of length-n weak ascent sequences with maximal number of weak ascents is A000108(n).
(End)
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LINKS
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EXAMPLE
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There are a(4) = 23 weak ascent sequences (dots for zeros):
1: [ . . . . ]
2: [ . . . 1 ]
3: [ . . . 2 ]
4: [ . . . 3 ]
5: [ . . 1 . ]
6: [ . . 1 1 ]
7: [ . . 1 2 ]
8: [ . . 1 3 ]
9: [ . . 2 . ]
10: [ . . 2 1 ]
11: [ . . 2 2 ]
12: [ . . 2 3 ]
13: [ . 1 . . ]
14: [ . 1 . 1 ]
15: [ . 1 . 2 ]
16: [ . 1 1 . ]
17: [ . 1 1 1 ]
18: [ . 1 1 2 ]
19: [ . 1 1 3 ]
20: [ . 1 2 . ]
21: [ . 1 2 1 ]
22: [ . 1 2 2 ]
23: [ . 1 2 3 ]
(End)
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0, 1,
add(b(n-1, j, t+`if`(j>=i, 1, 0)), j=0..t+1))
end:
a:= n-> b(n, -1$2):
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PROG
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(PARI) \\ see formula (5) on page 18 of the Benyi/Claesson/Dukes reference
N=40;
M=matrix(N, N, r, c, -1); \\ memoization
a(n, k)=
{
if ( n==0 && k==0, return(1) );
if ( k==0, return(0) );
if ( n==0, return(0) );
if ( M[n, k] != -1 , return( M[n, k] ) );
my( s );
s = sum( i=0, n, sum( j=0, k-1,
(-1)^j * binomial(k-j, i) * binomial(i, j) * a( n-i, k-j-1 )) );
M[n, k] = s;
return( s );
}
for (n=0, N, print1( sum(k=1, n, a(n, k)), ", "); );
\\ print triangle a(n, k), see A369321:
\\ for (n=0, N, for(k=0, n, print1(a(n, k), ", "); ); print(); );
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CROSSREFS
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Cf. A000079, A000108, A000110, A022493, A091768, A102038, A113227, A263777, A328441, A336071, A336072.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(0)=1 prepended and more terms from Joerg Arndt, Jan 20 2024
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STATUS
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approved
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