A329741 -id:A329741 - cp's OEIS frontend

A329766 Number of compositions of n whose run-lengths cover an initial interval of positive integers.

1, 1, 1, 3, 6, 13, 21, 48, 89, 180, 355, 707, 1382, 2758, 5448, 10786, 21391, 42476, 84291, 167516, 333036, 662153, 1317687, 2622706, 5221951, 10400350, 20720877, 41288823, 82294979, 164052035, 327088649, 652238016, 1300788712, 2594486045, 5175378128, 10324522020

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)  (3)    (4)      (5)
                (1,2)  (1,3)    (1,4)
                (2,1)  (3,1)    (2,3)
                       (1,1,2)  (3,2)
                       (1,2,1)  (4,1)
                       (2,1,1)  (1,1,3)
                                (1,2,2)
                                (1,3,1)
                                (2,1,2)
                                (2,2,1)
                                (3,1,1)
                                (1,1,2,1)
                                (1,2,1,1)

Looking at multiplicities instead of run-lengths gives A329741.
The complete case is A329749.
Complete compositions are A107429.
Cf. A000740, A008965, A242882, A244164, A274174, A329738, A329739, A329740, A329744, A329748.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[Length/@Split[#]]&]],{n,0,10}]

a(21)-a(26) from Giovanni Resta, Nov 22 2019

a(27)-a(35) from Alois P. Heinz, Jul 06 2020

A332340 Number of widely alternately co-strongly normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 9, 11, 13, 23, 53, 78, 120, 207, 357, 707, 1183, 2030, 3558, 6229, 10868

Offset: 0

Gus Wiseman, Feb 17 2020

An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (12)   (121)   (122)    (123)     (1213)     (1232)
             (21)   (211)   (212)    (132)     (1231)     (1322)
             (111)  (1111)  (1211)   (213)     (1312)     (2123)
                            (11111)  (231)     (1321)     (2132)
                                     (312)     (2122)     (2312)
                                     (321)     (2131)     (2321)
                                     (1212)    (2311)     (3122)
                                     (2121)    (3121)     (3212)
                                     (111111)  (3211)     (12131)
                                               (12121)    (13121)
                                               (1111111)  (21212)
                                                          (122111)
                                                          (11111111)
For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).

Normal compositions are A107429.
Compositions with normal run-lengths are A329766.
The Heinz numbers of the case of partitions are A332290.
The case of partitions is A332289.
The total (instead of alternating) version is A332337.
Not requiring normality gives A332338.
The strong version is this same sequence.
The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A317245, A317491, A329744, A329741, A329746, A332278, A332279, A332292.

totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]

A329740 Number of compositions of n whose multiplicities are distinct and cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 4, 10, 10, 10, 73, 196, 133, 379, 319, 379, 502, 805, 562, 1108, 13648, 51448, 51691, 115174, 140011, 178597, 203617, 329737, 292300, 456703, 456160, 608386, 633466, 898186, 823009, 39014392, 190352269, 266293795, 493345615, 834326995, 947714938

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(9) = 10 compositions:
  (1)  (2)  (3)  (4)      (5)      (6)      (7)      (8)      (9)
                 (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
                 (1,2,1)  (1,2,2)  (1,4,1)  (1,3,3)  (1,6,1)  (1,4,4)
                 (2,1,1)  (1,3,1)  (4,1,1)  (1,5,1)  (2,2,4)  (1,7,1)
                          (2,1,2)           (2,2,3)  (2,3,3)  (2,2,5)
                          (2,2,1)           (2,3,2)  (2,4,2)  (2,5,2)
                          (3,1,1)           (3,1,3)  (3,2,3)  (4,1,4)
                                            (3,2,2)  (3,3,2)  (4,4,1)
                                            (3,3,1)  (4,2,2)  (5,2,2)
                                            (5,1,1)  (6,1,1)  (7,1,1)

The version allowing repeated multiplicities is A329741.
Complete compositions are A107429.
Compositions whose multiplicities are distinct are A242882.
Cf. A059966, A098504, A244164, A274174, A329739, A329766, A329748.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Range[Length[Union[#]]]==Sort[Length/@Split[Sort[#]]]&]],{n,0,10}]

a(21)-a(40) from Alois P. Heinz, Nov 21 2019

A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.

Original entry on oeis.org

1, 1, 3, 6, 15, 23, 53, 94, 203, 404, 855, 1648, 3416, 6662, 13400, 26406, 53038, 105306, 212051, 422162, 849267, 1696864, 3406077, 6807024, 13642099, 27268122, 54576003, 109096436, 218250874, 436243705, 872533347, 1744312748, 3488432736, 6974783481

Offset: 1

Alois P. Heinz, Jun 21 2014

nonn

			From _Gus Wiseman_, Nov 25 2019: (Start)
The a(1) = 1 through a(5) = 15 compositions:
  (1)  (2)  (3)    (4)      (5)
            (1,2)  (1,3)    (1,4)
            (2,1)  (3,1)    (2,3)
                   (1,1,2)  (3,2)
                   (1,2,1)  (4,1)
                   (2,1,1)  (1,1,3)
                            (1,2,2)
                            (1,3,1)
                            (2,1,2)
                            (2,2,1)
                            (3,1,1)
                            (1,1,1,2)
                            (1,1,2,1)
                            (1,2,1,1)
                            (2,1,1,1)
(End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..2000 (first 400 terms from Alois P. Heinz)
Vaclav Kotesovec, Graph a(n)/2^n

Column k=1 of A242451.
The complement is counted by A240085.
Cf. A003242, A098504, A114901, A261983, A329740, A329741.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 1) -b(n$2, 0, 2):
seq(a(n), n=1..50);

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Min@@Length/@Split[Sort[#]]==1&]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

a(n) = 2^(n-1) - A240085(n). - Gus Wiseman, Nov 25 2019

A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 2, 3, 3, 6, 12, 12, 42, 114, 210, 60, 360, 720, 1320, 1590, 3690, 6450, 16110, 33120, 59940, 61320, 112980, 171780, 387240, 803880, 769440, 1773240, 2823240, 5790960, 9916200, 19502280, 28244160, 56881440, 130548600, 279578880, 320554080, 541323720

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.

			The a(1) = 1 through a(8) = 12 compositions (empty column not shown):
  (1)  (12)  (112)  (122)  (123)  (1123)  (1223)
       (21)  (121)  (212)  (132)  (1132)  (1232)
             (211)  (221)  (213)  (1213)  (1322)
                           (231)  (1231)  (2123)
                           (312)  (1312)  (2132)
                           (321)  (1321)  (2213)
                                  (2113)  (2231)
                                  (2131)  (2312)
                                  (2311)  (2321)
                                  (3112)  (3122)
                                  (3121)  (3212)
                                  (3211)  (3221)

Looking at run-lengths instead of multiplicities gives A329749.
The non-complete version is A329741.
Complete compositions are A107429.
Cf. A008965, A098504, A244164, A274174, A329740, A329744, A329766.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[Sort[#]]]&]],{n,0,10}]

a(21)-a(38) from Alois P. Heinz, Jul 06 2020

A335942 Number of compositions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

Original entry on oeis.org

1, 1, 2, 2, 3, 12, 12, 32, 51, 144, 191, 486, 679, 1487, 3149, 5909, 11637, 18630, 36928, 76431, 141009, 264784, 535057, 921105, 1774022, 3388054, 6303519, 12255373, 22527578, 43358822, 77695383, 145170435, 264722429, 527776034, 936538336, 1807344134

Offset: 0

Alois P. Heinz, Jun 30 2020

nonn

			a(4) = 3: 211, 121, 112.
a(5) = 12: 23, 32, 113, 122, 131, 212, 221, 311, 1112, 1121, 1211, 2111.

Chai Wah Wu, Table of n, a(n) for n = 0..142 (n = 0..100 from Alois P. Heinz)

Cf. A107429, A329741, A335941.

b:= proc(n, i, s, p) option remember;
     `if`(n=0, `if`(s={$0..max(s)}, p!, 0), `if`(i<1, 0, add(
      b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}, p+j)/j!, j=0..n/i)))
    end:
a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}, 0):
seq(a(n), n=0..35);

b[n_, i_, s_, p_] := b[n, i, s, p] =
     If[n == 0, If[s == Range[0, Max[s]], p!, 0], If[i < 1, 0, Sum[
     b[n - i*j, i - 1, Union@Flatten@{s, j, If[j == 0, 0, i]}, p + j]/j!,
     {j, 0, n/i}]]];
a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)

A329749 Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 2, 3, 5, 11, 23, 40, 80, 180, 344, 661, 1321, 2657, 5268, 10481, 20903, 41572, 82734, 164998, 328304, 654510, 1305421, 2598811, 5182174, 10332978, 20594318, 41066611, 81897091, 163309679, 325707492, 649648912, 1295827380, 2584941276, 5156774487

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.

			The a(0) = 1 through a(6) = 11 compositions (empty column not shown):
  ()  (1)  (1,2)  (1,1,2)  (1,2,2)    (1,2,3)
           (2,1)  (1,2,1)  (2,1,2)    (1,3,2)
                  (2,1,1)  (2,2,1)    (2,1,3)
                           (1,1,2,1)  (2,3,1)
                           (1,2,1,1)  (3,1,2)
                                      (3,2,1)
                                      (1,2,1,2)
                                      (1,2,2,1)
                                      (2,1,1,2)
                                      (2,1,2,1)
                                      (1,1,2,1,1)

Looking at  multiplicities instead of run-lengths gives A329748.
The non-complete version is A329766.
Complete compositions are A107429.
Cf. A000740, A008965, A098504, A244164, A274174, A329738, A329739, A329740, A329741, A329744.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[#]]&]],{n,0,10}]

a(21)-a(35) from Alois P. Heinz, Jul 06 2020

cp's OEIS Frontend

A329766 Number of compositions of n whose run-lengths cover an initial interval of positive integers.

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A332340 Number of widely alternately co-strongly normal compositions of n.

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A329740 Number of compositions of n whose multiplicities are distinct and cover an initial interval of positive integers.

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A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.

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A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

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A335942 Number of compositions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

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Maple

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A329749 Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.

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Mathematica

Extensions