A328409
Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i <= j >= k.
Original entry on oeis.org
1, 1, 2, 3, 6, 16, 57, 245, 1248, 7151, 46104, 325560, 2523437, 21106494, 190806861, 1842347541, 19018910502, 208088481921, 2414462433024, 29512737830802, 380156646308541, 5133381861786182, 72678441538790901, 1074324277172134786, 16581261996774703606
Offset: 0
a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 3: 000, 010, 011.
a(4) = 6: 0000, 0101, 0102, 0103, 0110, 0111.
a(5) = 16: 00000, 01010, 01011, 01020, 01021, 01022, 01030, 01031, 01032, 01033, 01101, 01102, 01103, 01104, 01110, 01111.
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b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`((i>j
xor t) and c=0, 0, b(n-1, i, is(i b(n, 0, true, 2):
seq(a(n), n=0..24);
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b[n_, j_, t_, c_] := b[n, j, t, c] = If[n == 0, 1, Sum[If[Xor[i>j, t] && c == 0, 0, b[n - 1, i, iJean-François Alcover, Feb 26 2020, after Alois P. Heinz *)
A328425
Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i < j > k or i >= j <= k.
Original entry on oeis.org
1, 1, 2, 4, 11, 36, 142, 647, 3383, 19816, 129162, 923279, 7201951, 60720996, 551268926, 5352973967, 55430433719, 609033864160, 7083303687843, 86864585123112, 1120997775904467, 15176639841694385, 215196709973260722, 3187766448289854016, 49262381105608795771
Offset: 0
a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 4: 000, 001, 002, 010.
a(5) = 11: 0000, 0001, 0002, 0003, 0010, 0020, 0021, 0100, 0101, 0102, 0103.
a(6) = 36: 00000, 00001, 00002, 00003, 00004, 00010, 00020, 00021, 00030, 00031, 00032, 00100, 00101, 00102, 00103, 00104, 00200, 00201, 00202, 00203, 00204, 00211, 00212, 00213, 00214, 01000, 01001, 01002, 01003, 01004, 01010, 01020, 01021, 01030, 01031, 01032.
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b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`((ij), max(0, c-1))), i=1..n))
end:
a:= n-> b(n, 0, true, 2):
seq(a(n), n=0..24);
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b[n_, j_, t_, c_] := b[n, j, t, c] = If[n == 0, 1, Sum[If[Xor[i < j, t] && c == 0, 0, b[n - 1, i, i > j, Max[0, c - 1]]], {i, 1, n}]];
a[n_] := b[n, 0, True, 2];
a /@ Range[0, 24] (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)
A326308
Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i < j > k.
Original entry on oeis.org
1, 1, 2, 1, 3, 6, 26, 85, 476, 2171, 14905, 87153, 708825, 5053464, 47514180, 399542814, 4264132468, 41306091312, 493337571005, 5408829555639, 71476985762027, 874870165668858, 12673922434134249, 171294209823727623, 2699365743596908540, 39925463781029750810
Offset: 0
a(6) = 26: 010101, 010102, 010103, 010104, 010105, 010201, 010202, 010203, 010204, 010205, 010212, 010213, 010214, 010215, 010301, 010302, 010303, 010304, 010305, 010312, 010313, 010314, 010315, 010323, 010324, 010325.
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b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add(
`if`(c>0 or i>j and t or ij), max(0, c-1)), 0), i=1..n))
end:
a:= n-> b(n, 0, true$2, 2):
seq(a(n), n=0..25);
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b[n_, j_, t_, u_, c_] := b[n, j, t, u, c] = If[n == 0, 1, Sum[If[c>0 || i>j && t || ij, Max[0, c-1]], 0], {i, 1, n}]];
a[n_] := b[n, 0, True, True, 2];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)
A326412
Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i >= j <= k or i <= j >= k.
Original entry on oeis.org
1, 1, 2, 5, 17, 69, 330, 1797, 11028, 74932, 559351, 4540088, 39840318, 375421225, 3782383945, 40548234374, 460956742449, 5536790753853, 70077462043662, 931945968071778, 12993337101354500, 189485727877247991, 2884989393948284323, 45772604755492432599
Offset: 0
a(4) = 17: 0000, 0001, 0002, 0003, 0010, 0011, 0020, 0021, 0022, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113.
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b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add(
`if`(c>0 or i>=j and t or i<=j and u, b(n-1, i,
is(i<=j), is(i>=j), max(0, c-1)), 0), i=1..n))
end:
a:= n-> b(n, 0, true$2, 2):
seq(a(n), n=0..25);
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b[n_, j_, t_, u_, c_] := b[n, j, t, u, c] = If[n == 0, 1, Sum[If[c > 0 || i >= j && t || i <= j && u, b[n - 1, i, i <= j, i >= j , Max[0, c - 1]], 0], {i, 1, n}]];
a[n_] := b[n, 0, True, True, 2];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)
Showing 1-4 of 4 results.