A097979 -id:A097979 - cp's OEIS frontend

A238341 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the largest part, n>=0, 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 1, 0, 1, 0, 12, 3, 0, 0, 1, 0, 23, 7, 1, 0, 0, 1, 0, 46, 13, 4, 0, 0, 0, 1, 0, 91, 25, 10, 1, 0, 0, 0, 1, 0, 183, 46, 21, 5, 0, 0, 0, 0, 1, 0, 367, 89, 39, 15, 1, 0, 0, 0, 0, 1, 0, 737, 175, 70, 35, 6, 0, 0, 0, 0, 0, 1, 0, 1478, 351, 125, 71, 21, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Columns k=0-10 give: A000007, A097979(n-1) for n>0, A243737, A243738, A243739, A243740, A243741, A243742, A243743, A243744, A243745.
T(n^2,n) gives A243746(n).
Row sums are A011782.

			Triangle starts:
00:  1;
01:  0,     1;
02:  0,     1,    1;
03:  0,     3,    0,   1;
04:  0,     6,    1,   0,   1;
05:  0,    12,    3,   0,   0,   1;
06:  0,    23,    7,   1,   0,   0,  1;
07:  0,    46,   13,   4,   0,   0,  0, 1;
08:  0,    91,   25,  10,   1,   0,  0, 0, 1;
09:  0,   183,   46,  21,   5,   0,  0, 0, 0, 1;
10:  0,   367,   89,  39,  15,   1,  0, 0, 0, 0, 1;
11:  0,   737,  175,  70,  35,   6,  0, 0, 0, 0, 0, 1;
12:  0,  1478,  351, 125,  71,  21,  1, 0, 0, 0, 0, 0, 1;
13:  0,  2962,  710, 229, 131,  56,  7, 0, 0, 0, 0, 0, 0, 1;
14:  0,  5928, 1443, 435, 230, 126, 28, 1, 0, 0, 0, 0, 0, 0, 1,
15:  0, 11858, 2926, 859, 395, 253, 84, 8, 0, 0, 0, 0, 0, 0, 0, 1;
...

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Cf. A026794 (the same for partitions), A238342 (the same for smallest part).

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_, k_] := Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]; a[0, 0] = 1; a[, 0] = 0; Table[a[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* _Jean-François Alcover, Jan 19 2015, after Maple code in A243737 *)

A105039 Number of compositions of n with unique smallest part.

Original entry on oeis.org

1, 1, 3, 3, 8, 11, 20, 34, 59, 96, 167, 282, 475, 800, 1352, 2275, 3828, 6426, 10785, 18085, 30297, 50698, 84770, 141623, 236425, 394381, 657380, 1094975, 1822628, 3031843, 5040129, 8373594, 13903588, 23072567, 38267330, 63435438, 105103059, 174054820, 288105394

Offset: 1

Vladeta Jovovic, Apr 03 2005

			a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Vincenzo Librandi)

Cf. A079501, A097979.

Column k=1 of A238342.

G:= sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..70): Gser:=series(G,x=0,44): seq(coeff(Gser,x^n),n=1..41); # Emeric Deutsch, Apr 13 2005

nn=37;Drop[CoefficientList[Series[Sum[x^j/(1-x^(j+1)/(1-x))^2,{j,1,nn}],{x,0,nn}],x],1] (* Geoffrey Critzer, Mar 31 2014 *)

a(n)=1+sum(k=2,(n+3)\2,k*sum(s=1,(n-1)\k,binomial(n-k*s-1,k-2))) (Alekseyev)

G.f.: Sum_{k>0} k * x^(2*k-1)/((1 - x^k) * (1 - x)^(k-1)) = (1 - x)^2 * Sum_{k>0} x^k/(1 - x - x^(k+1))^2.
a(n) = 1 + sum(k=2..[(n+3)/2], k * sum(s=1..[(n-1)/k], binomial(n-k*s-1, k-2) ) ). - Max Alekseyev, Apr 15 2005
a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 02 2014
Equivalently, a(n) ~ n * phi^(n-3) / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

More terms from Emeric Deutsch and Max Alekseyev, Apr 13 2005

A363224 Number of integer compositions of n in which the least part appears more than once.

Original entry on oeis.org

0, 1, 1, 5, 8, 21, 44, 94, 197, 416, 857, 1766, 3621, 7392, 15032, 30493, 61708, 124646, 251359, 506203, 1018279, 2046454, 4109534, 8246985, 16540791, 33160051, 66451484, 133122753, 266612828, 533839069, 1068701695, 2139110054, 4281063708, 8566862025

Offset: 1

Gus Wiseman, Jun 04 2023

nonn

Also the number of multisets of length n covering an initial interval of positive integers with more than one co-mode.

			The a(1) = 0 through a(6) = 21 compositions:
  .  (11)  (111)  (22)    (113)    (33)
                  (112)   (131)    (114)
                  (121)   (311)    (141)
                  (211)   (1112)   (222)
                  (1111)  (1121)   (411)
                          (1211)   (1113)
                          (2111)   (1122)
                          (11111)  (1131)
                                   (1212)
                                   (1221)
                                   (1311)
                                   (2112)
                                   (2121)
                                   (2211)
                                   (3111)
                                   (11112)
                                   (11121)
                                   (11211)
                                   (12111)
                                   (21111)
                                   (111111)

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

The complement is counted by A105039.
For partitions instead of compositions we have A117989.
Row sums of columns k > 1 of A238342.
If all parts appear more than once we have A240085, for partitions A007690.
If the least part appears exactly twice we have A241862.
For greatest instead of least we have A363262, see triangle A238341.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
Cf. A002865, A008284, A097979, A243737.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Min@@#]>1&]],{n,15}]

C_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0],Vec(h))}
C_x(35) \\ John Tyler Rascoe, Jul 06 2024

G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - John Tyler Rascoe, Jul 06 2024

A363262 Number of integer compositions of n in which the greatest part appears more than once.

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 18, 37, 73, 145, 287, 570, 1134, 2264, 4526, 9061, 18152, 36374, 72884, 146011, 292416, 585422, 1171632, 2344136, 4688821, 9376832, 18749169, 37485358, 74939850, 149813328, 299492966, 598729533, 1196987066, 2393137399, 4784846896, 9567357951

Offset: 1

Gus Wiseman, Jun 04 2023

nonn

Also the number of multisets of length n covering an initial interval of positive integers with more than one mode.

			The a(2) = 1 through a(6) = 9 compositions:
  (11)  (111)  (22)    (122)    (33)
               (1111)  (212)    (222)
                       (221)    (1122)
                       (11111)  (1212)
                                (1221)
                                (2112)
                                (2121)
                                (2211)
                                (111111)

For partitions instead of compositions we have A002865.
The complement is counted by A097979 shifted left.
Row sums of columns k > 1 of A238341.
If all parts appear more than once we have A240085, for partitions A007690.
If the greatest part appears exactly twice we have A243737.
For least instead of greatest we have A363224, see triangle A238342.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
Cf. A008284, A105039, A117989, A362607, A362608, A362612, A362614.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Max@@#]>1&]],{n,15}]

A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 4, 4, 5, 7, 10, 8, 13, 13, 15, 19, 25, 24, 35, 35, 43, 50, 61, 59, 79, 83, 98, 111, 137, 137, 176, 187, 219, 240, 284, 298, 360, 385, 444, 485, 568, 600, 706, 763, 867, 951, 1088, 1168, 1345, 1453, 1641, 1792, 2023, 2179, 2467, 2673, 2988

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			The a(1) = 1 through a(10) = 7 partitions:
  1  11  111  211   221    21111   2221     22211     22221      33211
              1111  2111   111111  22111    221111    32211      222211
                    11111          211111   2111111   2211111    322111
                                   1111111  11111111  21111111   2221111
                                                      111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(9) = 5 through a(12) = 8 partitions:
  (22221)      (33211)       (33221)        (2222211)
  (32211)      (222211)      (222221)       (3222111)
  (2211111)    (322111)      (322211)       (3321111)
  (21111111)   (2221111)     (332111)       (32211111)
  (111111111)  (22111111)    (2222111)      (222111111)
               (211111111)   (3221111)      (2211111111)
               (1111111111)  (22211111)     (21111111111)
                             (221111111)    (111111111111)
                             (2111111111)
                             (11111111111)

For parts instead of multiplicities we have A087897, complement A000009.
For multisets instead of partitions we have A105039, complement A363224.
The complement is counted by A363264.
For mode we have A363484, complement A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A025147, A096765, A117989, A243737, A362612.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]==1&]]],{n,0,30}]

A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 3, 2, 7, 5, 9, 12, 13, 13, 22, 19, 29, 33, 39, 43, 63, 63, 82, 94, 111, 119, 159, 164, 203, 229, 272, 301, 370, 400, 479, 538, 628, 692, 826, 904, 1053, 1181, 1353, 1502, 1742, 1919, 2205, 2456, 2790, 3097, 3539, 3911, 4435, 4929

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

For parts instead of multiplicities we have A000009, complement A087897.
For multisets instead of partitions we have A363224, complement A105039.
The complement is counted by A363263.
For mode we have A363485, complement A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A067029 counts minima in prime factorization, co-modes A362613.
A071178 counts maxima in prime factorization, modes A362611.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A096765, A117989, A243737, A275870, A362612.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]>1&]]],{n,0,30}]

A363484 Number of integer partitions of n covering an initial interval of positive integers with a unique mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 5, 6, 6, 8, 11, 12, 17, 20, 21, 27, 35, 38, 50, 56, 65, 76, 95, 105, 125, 146, 167, 198, 233, 252, 305, 351, 394, 457, 522, 585, 681, 778, 878, 994, 1135, 1269, 1446, 1638, 1828, 2067, 2339, 2613, 2940, 3301, 3684, 4143, 4634, 5156, 5771

Offset: 0

Gus Wiseman, Jun 05 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (211)   (221)    (21111)   (2221)     (3221)
                    (1111)  (2111)   (111111)  (3211)     (22211)
                            (11111)            (22111)    (32111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of multiplicities we have A096765, complement A025147.
For multisets instead of partitions we have A097979, complement A363262.
For co-mode we have A363263, complement A363264.
The complement is counted by A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]==1&]]],{n,0,30}]

A377823 Sum of the positions of maximum parts in all compositions of n.

Original entry on oeis.org

0, 1, 4, 10, 23, 50, 110, 240, 526, 1147, 2489, 5368, 11510, 24543, 52090, 110109, 231959, 487245, 1020980, 2134838, 4455582, 9283742, 19314740, 40128699, 83265342, 172564435, 357228078, 738707908, 1526004117, 3149310585, 6493394292, 13376521031, 27532616663

Offset: 0

John Tyler Rascoe, Nov 08 2024

			The composition of 7, (1,2,1,1,2) has maximum parts at positions 2 and 5; so it contributes 7 to a(7) = 240.

Cf. A001792, A010054, A011782, A097976, A097979, A377824.

A_xy(N) = {my(x='x+O('x^N), h = sum(i=1,N, y^(i*(i+1)/2)*x^i)+sum(m=2,N, sum(i=1,N, ((y^i)*x^m)*((x-x^m)/(1-x))^(i-1)*(sum(j=0,N-m-i, prod(u=1,j, (x-x^m)/(1-x)+(y^(u+i))*x^m)))))); h}
P_xy(N) = Pol(A_xy(N), {x})
A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}
A_x(20)

G.f.: A(x) = d/dy A(x,y)|{y = 1}, where A(x,y) = Sum{i>0} (x^i * y^(i*(i+1)/2)) + Sum_{m>1} (Sum_{i>0} (x^m * y^i * ((x-x^m)/(1-x))^(i-1) * (Sum_{j>=0} (Product_{u=1..j} ((x-x^m)/(1-x) + x^m * y^(u+i)) ) ) ) ).

A284942 Expansion of Sum_{k>=1} mu(k)^2x^k(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 8, 19, 46, 107, 244, 547, 1213, 2665, 5807, 12567, 27042, 57899, 123428, 262115, 554750, 1170538, 2463154, 5170462, 10829234, 22635087, 47223412, 98353299, 204519549, 424665001, 880581806, 1823667221, 3772341661, 7794697759, 16089424392, 33178906531, 68357928558

Offset: 1

Ilya Gutkovskiy, Apr 06 2017

nonn

Total number of squarefree parts in all compositions (ordered partitions) of n.

			a(4) = 19 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 0 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 19.

Alois P. Heinz, Table of n, a(n) for n = 1..3312
Index entries for sequences related to compositions

Cf. A005117, A008683, A011782, A059570, A097941, A097979, A102291, A281573, A284943.

a:= proc(n) option remember; add(`if`(numtheory[
      issqrfree](j), ceil(2^(n-j-1)), 0)+a(n-j), j=1..n)
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

nmax = 33; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k (1 - x)^2/(1 - 2 x)^2, {k, 1, nmax}], {x, 0, nmax}], x]]

x='x+O('x^34); Vec(sum(k=1, 34, moebius(k) ^2*x^k*(1 - x)^2/(1 - 2*x)^2)) \\ Indranil Ghosh, Apr 06 2017

G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2.

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)(1 - x)^2/(1 - 2x)^2.

Original entry on oeis.org

0, 1, 3, 8, 20, 47, 110, 251, 564, 1251, 2750, 5994, 12978, 27934, 59825, 127565, 270959, 573575, 1210466, 2547562, 5348385, 11203292, 23419629, 48865346, 101782870, 211670094, 439548898, 911515214, 1887865266, 3905400206, 8070139762, 16658958223, 34355273843

Offset: 1

Ilya Gutkovskiy, Apr 06 2017

nonn

Total number of prime power parts (1 excluded) in all compositions (ordered partitions) of n.

			a(5) = 20 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 4], [1, 3, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 3], [1, 1, 2, 1], [1, 1, 1, 2], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 0 = 20.

Alois P. Heinz, Table of n, a(n) for n = 1..3313
Index entries for sequences related to compositions

Cf. A011782, A059570, A073335, A097941, A097979, A102291, A246655, A284942.

b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

nmax = 33; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[k]] x^k (1 - x)^2/(1 - 2 x)^2, {k, 2, nmax}], {x, 0, nmax}], x]]

x='x+O('x^34); concat([0], Vec(sum(k=2, 34, (1\omega(k))*x^k*(1 - x)^2/(1 - 2*x)^2))) \\ Indranil Ghosh, Apr 06 2017

G.f.: Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

cp's OEIS Frontend

A238341 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the largest part, n>=0, 0<=k<=n.

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A105039 Number of compositions of n with unique smallest part.

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A363224 Number of integer compositions of n in which the least part appears more than once.

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A363262 Number of integer compositions of n in which the greatest part appears more than once.

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A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

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A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

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A363484 Number of integer partitions of n covering an initial interval of positive integers with a unique mode.

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Mathematica

A377823 Sum of the positions of maximum parts in all compositions of n.

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A284942 Expansion of Sum_{k>=1} mu(k)^2x^k(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)(1 - x)^2/(1 - 2x)^2.