A320981
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of seven.
Original entry on oeis.org
0, 1, 8, 1031, 9379, 486299, 5084162, 196352061, 2352460536, 81070646577, 1103813259377, 36592927821767, 560827842703887, 18549898652794829, 317078625531545481, 10667284286197389079, 201655098112826170280, 6973904694490809821089, 144705018721890264334923
Offset: 7
-
b:= proc(u, o, c) option remember; `if`(c<0 or c>7, 0, `if`(u+o=0,
x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..7))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 7):
seq(a(n), n=7..30);
A320982
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of eight.
Original entry on oeis.org
0, 1, 9, 2061, 20736, 1507293, 17118482, 849597609, 10896848134, 459634150710, 6622740258098, 260269850813548, 4183025733598532, 160312786582715100, 2852747036627655519, 109305622312149855474, 2138487419350175956196, 83104784124134380097202
Offset: 8
-
b:= proc(u, o, c) option remember; `if`(c<0 or c>8, 0, `if`(u+o=0,
x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..8))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 8):
seq(a(n), n=8..30);
A320983
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of nine.
Original entry on oeis.org
0, 1, 10, 4116, 45431, 4619654, 56724741, 3604309466, 49361628592, 2539183745028, 38672184949164, 1795814647499288, 30258670886635123, 1339822976545126729, 24829628782352186543, 1080751193228180109916, 21899879370157544611142, 954090805109110152360526
Offset: 9
-
b:= proc(u, o, c) option remember; `if`(c<0 or c>9, 0, `if`(u+o=0,
x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..9))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 9):
seq(a(n), n=9..30);
A320984
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of ten.
Original entry on oeis.org
0, 1, 11, 8220, 98827, 14052474, 185726938, 15068990276, 219771532102, 13747965316306, 220995092995233, 12094333663749818, 213504732754725133, 10896908531544406283, 210288454755592374452, 10376111829436767498058, 217839166931637914375624
Offset: 10
-
b:= proc(u, o, c) option remember; `if`(c<0 or c>10, 0, `if`(u+o=0,
x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..10))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 10):
seq(a(n), n=10..30);