A327244 -id:A327244 - cp's OEIS frontend

A032020 Number of compositions (ordered partitions) of n into distinct parts.

1, 1, 1, 3, 3, 5, 11, 13, 19, 27, 57, 65, 101, 133, 193, 351, 435, 617, 851, 1177, 1555, 2751, 3297, 4757, 6293, 8761, 11305, 15603, 24315, 30461, 41867, 55741, 74875, 98043, 130809, 168425, 257405, 315973, 431065, 558327, 751491, 958265, 1277867, 1621273

Offset: 0

Christian G. Bower, Apr 01 1998

Compositions into distinct parts are equivalent to (1,1)-avoiding compositions. - Gus Wiseman, Jun 25 2020

All terms are odd. - Alois P. Heinz, Apr 09 2021

			a(6) = 11 because 6 = 5+1 = 4+2 = 3+2+1 = 3+1+2 = 2+4 = 2+3+1 = 2+1+3 = 1+5 = 1+3+2 = 1+2+3.
From _Gus Wiseman_, Jun 25 2020: (Start)
The a(0) = 1 through a(7) = 13 strict compositions:
  ()  (1)  (2)  (3)    (4)    (5)    (6)      (7)
                (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
                (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                              (3,2)  (4,2)    (3,4)
                              (4,1)  (5,1)    (4,3)
                                     (1,2,3)  (5,2)
                                     (1,3,2)  (6,1)
                                     (2,1,3)  (1,2,4)
                                     (2,3,1)  (1,4,2)
                                     (3,1,2)  (2,1,4)
                                     (3,2,1)  (2,4,1)
                                              (4,1,2)
                                              (4,2,1)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
C. G. Bower, Transforms (2)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97. (free access)
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A008289, A032011.
Row sums of A241719.
Column k=1 of A243081, A242447, A261835, A261836, A327244, A327245.
Main diagonal of A261960.
Dominated by A003242 (anti-run compositions).
These compositions are ranked by A233564.
(1,1)-avoiding patterns are counted by A000142.
Numbers with strict prime signature are A130091.
(1,1,1)-avoiding compositions are counted by A232432.
(1,1)-matching compositions are counted by A261982.
Inseparable partitions are counted by A325535.
Patterns matched by compositions are counted by A335456.
Strict permutations of prime indices are counted by A335489.
Cf. A000009, A011782, A102726, A181796, A261962, A333489, A335457.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0))) end:
a:= proc(n) local l; l:=b(n, n): add((i-1)! *l[i], i=1..nops(l)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 12 2012
# second Maple program:
T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
    end:
a:= n-> add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 04 2015

f[list_]:=Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 0,30}]
T[n_, k_] := T[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], T[n-k, k] + k*T[n-k, k-1]]]; a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[8*n + 1] - 1) / 2]}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

N=66;  q='q+O('q^N);
gf=sum(n=0,N, n!*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf)
/* Joerg Arndt, Oct 20 2012 */

Q(N) = { \\ A008289
  my(q = vector(N)); q[1] = [1, 0, 0, 0];
  for (n = 2, N,
    my(m = (sqrtint(8*n+1) - 1)\2);
    q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
    for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
  return(q);
};
seq(N) = concat(1, apply(q -> sum(k = 1, #q, q[k] * k!), Q(N)));
seq(43) \\ Gheorghe Coserea, Sep 09 2018

"AGK" (ordered, elements, unlabeled) transform of 1, 1, 1, 1, ...
G.f.: Sum_{k>=0} k! * x^((k^2+k)/2) / Product_{j=1..k} (1-x^j). - David W. Wilson May 04 2000
a(n) = Sum_{m=1..n} A008289(n,m)*m!. - Geoffrey Critzer, Sep 07 2012

A327245 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 3, 10, 13, 0, 3, 39, 87, 75, 0, 5, 100, 510, 836, 541, 0, 11, 303, 2272, 7042, 9025, 4683, 0, 13, 782, 9999, 46628, 104255, 109110, 47293, 0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835, 0, 27, 5388, 154038, 1577256, 7676830, 19798096, 28538496, 21713032, 7087261

Offset: 0

Alois P. Heinz, Sep 14 2019

			T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  3,   10,    13;
  0,  3,   39,    87,     75;
  0,  5,  100,   510,    836,    541;
  0, 11,  303,  2272,   7042,   9025,    4683;
  0, 13,  782,  9999,  46628, 104255,  109110,   47293;
  0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-2 give: A000007, A032020 (for n>0), A327847.
Main diagonal gives A000670.
Row sums give A321586.
T(2n,n) gives A327589.
Cf. A327244, A327588.

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p + j] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327588(n).

A120774 Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 1, 2, 8, 31, 147, 899, 5777, 41024, 322488, 2749325, 25118777, 245389896, 2554780438, 28009868787, 323746545433, 3933023224691, 49924332801387, 661988844566017, 9138403573970063, 131043199040556235, 1949750421507432009, 30031656711776544610

Offset: 0

Alford Arnold, Jul 12 2006

Old name was: Row sums of A179233.

a(n) is the number of ways to linearly order the blocks in each set partition of {1,2,...,n} where two blocks are considered identical if they have the same number of elements. - Geoffrey Critzer, Sep 29 2011

			A179233 begins 1; 1; 1 1; 6 1 1; 8 3 18 1 1 ... with row sums 1, 1 2 8 31 147 ...
a(3) = 8: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3, 1|2|3. - _Alois P. Heinz_, Apr 27 2017

Alois P. Heinz, Table of n, a(n) for n = 0..525

Cf. A000041, A032011, A049019, A096161, A096162, A196301.
Row sums of A179233, A285824.
Main diagonal of A327244.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      (p+n)!/n!, add(b(n-i*j, i-1, p+j)*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 27 2017

f[{x_,y_}]:= x!^y y!;   Table[Total[Table[n!,{PartitionsP[n]}]/Apply[Times,Map[f,Map[Tally,Partitions[n]],{2}],2] * Apply[Multinomial,Map[Last,Map[Tally,Partitions[n]],{2}],2]],{n,0,20}]  (* Geoffrey Critzer, Sep 29 2011 *)

Leading 1 inserted, definition simplified by R. J. Mathar, Sep 28 2011

a(15) corrected, more terms, and new name (using Geoffrey Critzer's comment) from Alois P. Heinz, Apr 27 2017

A327673 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 3, 18, 19, 0, 3, 60, 171, 121, 0, 5, 210, 1173, 1996, 1041, 0, 11, 798, 7512, 22784, 27225, 11191, 0, 13, 2462, 39708, 196904, 411115, 382086, 130663, 0, 19, 7891, 204987, 1546042, 4991815, 7843848, 5932843, 1731969

Offset: 0

Alois P. Heinz, Sep 21 2019

			T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 18: 3aab, 3aba, 3baa, 3abb, 3bab, 3bba, 2aa1b, 2ab1a, 2ba1a, 2ab1b, 2ba1b, 2bb1a, 1a2ab, 1a2ba, 1a2bb, 1b2aa, 1b2ab, 1b2ba.
T(3,3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  3,   18,    19;
  0,  3,   60,   171,    121;
  0,  5,  210,  1173,   1996,   1041;
  0, 11,  798,  7512,  22784,  27225,  11191;
  0, 13, 2462, 39708, 196904, 411115, 382086, 130663;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-2 give: A000007, A032020 (for n>0), A327768.
Main diagonal gives A327674.
Row sums give A327675.
T(2n,n) gives A327678.
Cf. A327244, A327676.

b:= proc(n, i, k, p) option remember;
     `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, k_, p_] := b[n, i, k, p] = If[n==0, p!, If[i<1, 0, Sum[Binomial[ k^i, j] b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 30 2020, after Maple *)

Sum_{k=1..n} k * T(n,k) = A327676(n).

A309670 Number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

Original entry on oeis.org

1, 1, 3, 21, 115, 813, 7627, 71173, 740023, 8544169, 107195083, 1434581205, 20499413667, 312262663989, 4992164670007, 84221279919193, 1492818584618099, 27607818180267269, 533522844488072987, 10724970103003953053, 223859943086157531063, 4847766598150865273721

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Row sums of A327244.

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 05 2022, after Alois P. Heinz *)

A327595 Total number of colors in all colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

Original entry on oeis.org

0, 1, 5, 47, 343, 2989, 33185, 360963, 4279363, 55461897, 771543693, 11345355815, 176710558327, 2913914537349, 50149603855065, 906096874764227, 17125269159665511, 336432862441344121, 6882511824853124773, 146018382159954093023, 3207861915702573763355

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A327244.

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(k*b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
    b[n-i*j, Min[n-i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[k*b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A327244(n,k).

A327596 Number of colored compositions of 2n using all colors of an n-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

Original entry on oeis.org

1, 1, 27, 1222, 78819, 7990555, 1075539168, 185948116920, 39826324710186, 10231314625984628, 3097070454570888110, 1088018981038197792790, 436918864329884469153204, 198400793333371519398942287, 100941775818744369615731919906, 57064609834208008799145534143376

Offset: 0

Alois P. Heinz, Sep 18 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A327244.

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, i, 0)*(-1)^(n-i)*C(n, i), i=0..n):
seq(a(n), n=0..17);

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
    b[n-i*j, Min[n-i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, i, 0]*(-1)^(n-i)*c[n, i], {i, 0, n}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

a(n) = A327244(2n,n).

A327841 Number of colored compositions of n using all colors of a 2-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

Original entry on oeis.org

0, 0, 2, 10, 27, 70, 223, 508, 1193, 2822, 7048, 15690, 35072, 79018, 167667, 382976, 823599, 1742082, 3765187, 7785290, 16299074, 34337380, 70503188, 143916326, 296390373, 597048414, 1202172962, 2416614660, 4813022691, 9551780272, 18833189269, 37248671816

Offset: 0

Alois P. Heinz, Sep 27 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=2 of A327244.

b:= proc(n, i, k, p) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k, p+j)/
       j!*binomial(binomial(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> (k-> add(b(n$2, i, 0)*(-1)^(k-i)*
         binomial(k, i), i=0..k))(2):
seq(a(n), n=0..35);

cp's OEIS Frontend

A032020 Number of compositions (ordered partitions) of n into distinct parts.

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A327245 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A120774 Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.

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A327673 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A309670 Number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

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A327595 Total number of colors in all colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

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A327596 Number of colored compositions of 2n using all colors of an n-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

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A327841 Number of colored compositions of n using all colors of a 2-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.