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Search: a261981 -id:a261981
Displaying 1-5 of 5 results found.
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A261982
Number of compositions of n with some part repeated.
+10
55
0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
FORMULA
a(n) = A011782(n) - A032020(n).
G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020
EXAMPLE
a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.
MAPLE
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n], Length[#]>Length[Split[#]]&]], {n, 0, 10}] (* Gus Wiseman, Jun 24 2020 *)
CROSSREFS
Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 07 2015
STATUS
approved
A261983
Number of compositions of n such that at least two adjacent parts are equal.
+10
49
0, 0, 1, 1, 4, 9, 18, 41, 89, 185, 388, 810, 1670, 3435, 7040, 14360, 29226, 59347, 120229, 243166, 491086, 990446, 1995410, 4016259, 8076960, 16231746, 32599774, 65437945, 131293192, 263316897, 527912140, 1058061751, 2120039885, 4246934012, 8505864640
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 08 2015
a(n) = A011782(n) - A003242(n). - Emeric Deutsch, Jul 03 2020
EXAMPLE
a(5) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.
From Gus Wiseman, Jul 07 2020: (Start)
The a(2) = 1 through a(6) = 18 compositions:
(1,1) (1,1,1) (2,2) (1,1,3) (3,3)
(1,1,2) (1,2,2) (1,1,4)
(2,1,1) (2,2,1) (2,2,2)
(1,1,1,1) (3,1,1) (4,1,1)
(1,1,1,2) (1,1,1,3)
(1,1,2,1) (1,1,2,2)
(1,2,1,1) (1,1,3,1)
(2,1,1,1) (1,2,2,1)
(1,1,1,1,1) (1,3,1,1)
(2,1,1,2)
(2,2,1,1)
(3,1,1,1)
(1,1,1,1,2)
(1,1,1,2,1)
(1,1,2,1,1)
(1,2,1,1,1)
(2,1,1,1,1)
(1,1,1,1,1,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 0, add(
`if`(i=j, ceil(2^(n-j-1)), b(n-j, j)), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, x_, x_, ___}]&]], {n, 0, 10}] (* Gus Wiseman, Jul 06 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 0, Sum[If[i == j, Ceiling[2^(n-j-1)], b[n-j, j]], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz's Maple code *)
CROSSREFS
Column k=1 of A261981.
Cf. A011782, A262046.
The complement A003242 counts anti-runs.
Sum of positive-indexed terms of row n of A106356.
Row sums of A131044.
The (1,1,1) matching case is A335464.
Strict compositions are A032020.
Compositions with adjacent parts coprime are A167606.
Compositions with equal parts contiguous are A274174.
Cf. A114901, A178470, A242882, A244164, A325534, A335448, A335452.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 07 2015
STATUS
approved
A262191
Number T(n,k) of compositions of n such that k is the maximal distance between two identical parts; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows.
+10
8
1, 0, 1, 3, 1, 1, 4, 4, 2, 1, 5, 6, 6, 3, 1, 12, 13, 12, 9, 4, 1, 21, 23, 25, 21, 13, 5, 1, 36, 42, 46, 46, 34, 18, 6, 1, 43, 68, 88, 92, 80, 52, 24, 7, 1, 88, 119, 152, 180, 172, 132, 76, 31, 8, 1, 133, 197, 267, 330, 352, 304, 208, 107, 39, 9, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,4
LINKS
Alois P. Heinz, Rows n = 2..20, flattened
EXAMPLE
T(6,1) = 5: 33, 114, 411, 1122, 2211.
T(6,2) = 6: 141, 222, 1113, 1212, 2121, 3111.
T(6,3) = 6: 1131, 1221, 1311, 2112, 11112, 21111.
T(6,4) = 3: 11121, 11211, 12111.
T(6,5) = 1: 111111.
Triangle T(n,k) begins:
n\k: 1 2 3 4 5 6 7 8 9 10 11
---+----------------------------------------------------
02 : 1;
03 : 0, 1;
04 : 3, 1, 1;
05 : 4, 4, 2, 1;
06 : 5, 6, 6, 3, 1;
07 : 12, 13, 12, 9, 4, 1;
08 : 21, 23, 25, 21, 13, 5, 1;
09 : 36, 42, 46, 46, 34, 18, 6, 1;
10 : 43, 68, 88, 92, 80, 52, 24, 7, 1;
11 : 88, 119, 152, 180, 172, 132, 76, 31, 8, 1;
12 : 133, 197, 267, 330, 352, 304, 208, 107, 39, 9, 1;
MAPLE
b:= proc(n, s, l) option remember; `if`(n=0, 1, add(
`if`(j in s, 0, b(n-j, s union {`if`(l=[], j, l[1])},
`if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
end:
T:= (n, k)-> b(n, {}, [0$k]) -b(n, {}, [0$(k-1)]):
seq(seq(T(n, k), k=1..n-1), n=2..14);
MATHEMATICA
b[n_, s_, l_] := b[n, s, l] = If[n==0, 1, Sum[If[MemberQ[s, j], 0, b[n-j, s ~Union~ {If[l=={}, j, l[[1]]]}, If[l=={}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; T[n_, k_] := b[n, {}, Array[0&, k]] - b[n, {}, Array[0&, k-1]]; Table[T[n, k], {n, 2, 14}, { k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
CROSSREFS
Column k=1-5 gives A262192, A262194, A262196, A262197, A262200.
Row sums give A261982.
Cf. A261981.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 14 2015
STATUS
approved
A261960
Number A(n,k) of compositions of n such that no part equals any of its k immediate predecessors; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+10
6
1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 8, 1, 1, 1, 3, 4, 16, 1, 1, 1, 3, 3, 7, 32, 1, 1, 1, 3, 3, 5, 14, 64, 1, 1, 1, 3, 3, 5, 11, 23, 128, 1, 1, 1, 3, 3, 5, 11, 15, 39, 256, 1, 1, 1, 3, 3, 5, 11, 13, 23, 71, 512, 1, 1, 1, 3, 3, 5, 11, 13, 19, 37, 124, 1024
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Alois P. Heinz, Antidiagonals n = 0..50, flattened
EXAMPLE
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, 1, ...
: 2, 1, 1, 1, 1, 1, 1, ...
: 4, 3, 3, 3, 3, 3, 3, ...
: 8, 4, 3, 3, 3, 3, 3, ...
: 16, 7, 5, 5, 5, 5, 5, ...
: 32, 14, 11, 11, 11, 11, 11, ...
MAPLE
b:= proc(n, l) option remember;
`if`(n=0, 1, add(`if`(j in l, 0, b(n-j,
`if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
end:
A:= (n, k)-> b(n, [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[n_, l_] := b[n, l] = If[n==0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; A[n_, k_] := b[n, Array[0&, Min[n, k]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
CROSSREFS
Columns k=0-2 give: A011782, A003242, A261962.
Main diagonal gives A032020.
Cf. A261959, A261981.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 06 2015
STATUS
approved
A261984
Number of compositions of n such that the minimal distance between two identical parts equals two.
+10
3
0, 0, 0, 0, 1, 2, 3, 8, 16, 34, 57, 113, 213, 396, 733, 1333, 2419, 4400, 7934, 14321, 25687, 45947, 82085, 146410, 260547, 463021, 821669, 1456296, 2578051, 4559972, 8057373, 14225124, 25096606, 44246087, 77958821, 137283534, 241626535, 425079358, 747501363
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
a(n) ~ A003242(n). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
a(4) = 1: 121.
a(5) = 2: 131, 212.
a(6) = 3: 141, 1212, 2121.
a(7) = 8: 151, 232, 313, 1213, 1312, 2131, 3121, 12121.
a(8) = 16: 161, 242, 323, 1214, 1232, 1313, 1412, 2123, 2141, 2321, 3131, 3212, 4121, 12131, 13121, 21212.
MAPLE
g:= proc(n, i) option remember; `if`(n=0, 1, add(
`if`(i=j, 0, g(n-j, j)), j=1..n))
end:
b:= proc(n, i, m) option remember; `if`(n=0, 0, add(
`if`(i=j, 0, `if`(j=m, g(n-j, j), b(n-j, j, i))), j=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..45);
MATHEMATICA
g[n_, i_] := g[n, i] = If[n==0, 1, Sum[If[i==j, 0, g[n-j, j]], {j, 1, n}]];
b[n_, i_, m_] := b[n, i, m] = If[n==0, 0, Sum[If[i==j, 0, If[j==m, g[n-j, j], b[n-j, j, i]]], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 45] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)
CROSSREFS
Column k=2 of A261981.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 07 2015
STATUS
approved
page 1

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