A083710 -id:A083710 - cp's OEIS frontend

A168019 Square array A(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k+1.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 7, 2, 1, 0, 0, 1, 11, 0, 0, 0, 0, 0, 1, 15, 3, 0, 1, 0, 0, 0, 1, 22, 0, 2, 0, 0, 0, 0, 0, 1, 30, 5, 0, 0, 1, 0, 0, 0, 0, 1, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 56, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Omar E. Pol, Nov 21 2009

Note that column k lists each partition number A000041 followed by k zeros. See also A168020 and A168021.

Let A(n,k) denote the number of partitions of n into parts divisible by k+1. Let p(n) denote the number of partitions of n. If k+1 is a divisor of n then A(n,k) = p(n/(k+1)) otherwise A(n,k) = 0. [Conjectured by Omar E. Pol, Nov 25 2009] - this is trivial, just divide each part size by k - Franklin T. Adams-Watters, May 14 2010.

			The array, A(n, k), begins:
==================================================
... Column k: 0.. 1. 2. 3. 4. 5. 6. 7. 8. 9 10 11
. Row ...........................................
...n ............................................
==================================================
.. 0 ........ 1,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
.. 1 ........ 1,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.. 2 ........ 2,  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.. 3 ........ 3,  0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.. 4 ........ 5,  2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
.. 5 ........ 7,  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
.. 6 ....... 11,  3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0,
.. 7 ....... 15,  0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
.. 8 ....... 22,  5, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0,
.. 9 ....... 30,  0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0,
. 10 ....... 42,  7, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0,
. 11 ....... 56,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
. 12 ....... 77, 11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
...
Antidiagonal triangle, T(n, k), begins as:
   1;
   1, 1;
   2, 0, 1;
   3, 1, 0, 1;
   5, 0, 0, 0, 1;
   7, 2, 1, 0, 0, 1;
  11, 0, 0, 0, 0, 0, 1;
  15, 3, 0, 1, 0, 0, 0, 1;
  22, 0, 2, 0, 0, 0, 0, 0, 1;
  30, 5, 0, 0, 1, 0, 0, 0, 0, 1;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A000041, A035363, A083710, A135010.

Cf. A168016, A168120, A168021, A168121.

T[n_, k_]:= If[IntegerQ[(n-k)/(k+1)], PartitionsP[(n-k)/(k+1)], 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 13 2023 *)

def A168019(n,k): return number_of_partitions((n-k)/(k+1)) if ((n-k)%(k+1))==0 else 0
flatten([[A168019(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 13 2023

From G. C. Greubel, Jan 13 2023: (Start)
A(n, k) = A000041(n/(k+1)) if (k+1)|n, otherwise 0 (array).
T(n, k) = A000041((n-k)/(k+1)) if (k+1)|(n-k), otherwise 0 (antidiagonals).
A(n, 0) = T(n, 0) = A000041(n).
T(2*n, n) = A(n, n) = A000007(n).
Sum_{k=0..n} T(n, k) = A083710(n+1). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

Edited by Franklin T. Adams-Watters, May 14 2010

A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 5, 2, 5, 5, 4, 5, 7, 3, 5, 6, 5, 5, 9, 4, 7, 6, 6, 9, 11, 6, 9, 10, 9, 10, 12, 6, 11, 12, 10, 12, 16, 9, 15, 16, 12, 14, 18, 14, 16, 18, 14, 15, 22, 11, 16, 20, 13, 21, 23, 15, 21, 24, 21, 21, 31, 14, 24

Offset: 0

Gus Wiseman, Apr 19 2021

nonn

Alternative name: Number of strict integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

			The a(n) partitions for n = 14, 12, 18, 24, 30, 39, 36:
  (14)     (12)    (18)      (24)        (30)        (39)          (36)
  (12,2)   (8,4)   (12,6)    (16,8)      (24,6)      (36,3)        (27,9)
  (8,4,2)  (9,3)   (15,3)    (18,6)      (25,5)      (26,13)       (30,6)
           (10,2)  (16,2)    (20,4)      (27,3)      (27,9,3)      (32,4)
                   (12,4,2)  (21,3)      (28,2)      (28,7,4)      (33,3)
                             (22,2)      (20,10)     (30,6,3)      (34,2)
                             (12,6,4,2)  (18,9,3)    (24,12,3)     (24,12)
                                         (24,4,2)    (24,8,4,3)    (24,8,4)
                                         (16,8,4,2)  (20,10,5,4)   (18,9,6,3)
                                                     (24,6,4,3,2)  (24,6,4,2)
                                                                   (20,10,4,2)

The dual version is A098965 (non-strict: A083711).
The non-strict version is A339619 (Heinz numbers: complement of A343337).
The version with 1's allowed is A343347 (non-strict: A130689).
The case without a part dividing all the other parts is A343380.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A083710, A097986, A098743, A200745, A264401, A339563, A341450, A343337, A343341, A343344, A343378.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]

A137587 Triangle read by rows: A051731 * A026794.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 6, 1, 0, 0, 1, 11, 3, 2, 0, 0, 1, 12, 2, 1, 0, 0, 0, 1, 20, 6, 1, 2, 0, 0, 0, 1, 25, 4, 3, 1, 0, 0, 0, 0, 1, 37, 9, 2, 1, 2, 0, 0, 0, 0, 1, 43, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 70, 16, 6, 3, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 27 2008

That is, regard A051731 and A026794 as lower triangular square matrices and multiply them, then take the lower triangle of the product,
Left column = A083710 starting (1, 2, 3, 5, 6, 11, 12, ...).
Row sums = A047968.

			First few rows of the triangle:
   1;
   2, 1;
   3, 0, 1;
   5, 2, 0, 1;
   6, 1, 0, 0, 1;
  11, 3, 2, 0, 0, 1;
  12, 2, 1, 0, 0, 0, 1;
  20, 6, 1, 2, 0, 0, 0, 1;
  25, 4, 3, 1, 0, 0, 0, 0, 1;
  ...

Cf. A026794, A051731, A083710, A047968.

Inverse mobius transform of the partition triangle, A026794.

Typo in 9th row corrected by M. F. Hasler, Jun 08 2009

A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 0, 5, 2, 1, 0, 5, 3, 1, 0, 1, 7, 3, 1, 1, 0, 8, 4, 2, 1, 0, 10, 5, 2, 1, 0, 12, 6, 3, 1, 0, 15, 7, 3, 1, 0, 1, 17, 9, 4, 1, 1, 0, 21, 10, 4, 2, 1, 0, 25, 12, 6, 2, 1, 0, 29, 15, 6, 3, 1, 0, 35, 17, 8, 3, 1, 0

Offset: 0

Gus Wiseman, Apr 18 2021

The least gap (or mex) of a partition is the least positive integer that is not a part.

Row lengths are chosen to be consistent with the non-strict case A264401.

			Triangle begins:
   1
   0   1
   1   0
   1   0   1
   1   1   0
   2   1   0
   2   1   0   1
   3   1   1   0
   3   2   1   0
   5   2   1   0
   5   3   1   0   1
   7   3   1   1   0
   8   4   2   1   0
  10   5   2   1   0
  12   6   3   1   0
  15   7   3   1   0   1

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.

Row sums are A000009.
Row lengths are A002024.
Column k = 1 is A025147.
Column k = 2 is A025148.
The non-strict version is A264401.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A257993 gives the least gap of the partition with Heinz number n.
A339564 counts factorizations with a selected factor.
A342050 ranks partitions with even least gap.
A342051 ranks partitions with odd least gap.
Cf. A003242, A083710, A083711, A097986, A098743, A098965, A130689, A200745, A341450, A343347, A343377.

mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}]

A130708 Number of compositions of n such that every part divides the largest part.

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 26, 45, 79, 137, 241, 423, 754, 1343, 2410, 4344, 7870, 14305, 26103, 47763, 87649, 161229, 297251, 549108, 1016243, 1883898, 3497761, 6503420, 12107958, 22570221, 42121298, 78692765, 147165225, 275476533, 516115940

Offset: 0

Vladeta Jovovic, Jul 01 2007

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

Cf. A100346, A018818, A083710, A097986, A117086.

A130708 := proc(n) local gf,den1,den2,i,d ; gf := 1 ; for i from 1 to n do den1 := 1 ; den2 := 1 ; for d in numtheory[divisors](i) do den1 := den1-x^d ; if d < i then den2 := den2-x^d ; fi ; od ; gf := taylor(gf+x^i/den1/den2,x=0,n+1) ; od: coeftayl(gf,x=0,n) ; end: seq(A130708(n),n=0..40) ; # R. J. Mathar, Oct 28 2007

m = 35;
1 + Sum[x^n/((1 - Sum[x^d, {d, Divisors[n]}]) (1 - Sum[Boole[d < n] x^d, {d, Divisors[n]}])), {n, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, May 22 2020 *)

G.f.: 1 + Sum_{n>0} x^n/((1-Sum_{d divides n} x^d)*(1-Sum_{d divides n,d

More terms from R. J. Mathar, Oct 28 2007

A130711 Number of compositions of n such that the smallest part divides every part.

Original entry on oeis.org

1, 2, 4, 8, 14, 32, 57, 123, 239, 493, 970, 1997, 3953, 8017, 16024, 32281, 64550, 129742, 259561, 520606, 1041871, 2087177, 4176594, 8362063, 16730862, 33483361, 66987710, 134029333, 268117646, 536373213, 1072909785, 2146169660

Offset: 1

Vladeta Jovovic, Jul 01 2007

			a(5)=14 because among the 16 compositions of 5 only 2+3 and 3+2 do not qualify; the others, except for the composition 5, have at least one component equal to 1.

Cf. A083710.

G:=sum(x^n*(1-x^n)^2/((1-2*x^n)*(1-x^n-x^(2*n))), n=1..50); Gser:=series(G, x =0,40): seq(coeff(Gser,x,n),n=1..33); # Emeric Deutsch, Sep 08 2007

Inverse Moebius transform of A099036.

G.f.: Sum_{n>0} x^n*(1-x^n)^2/((1-2*x^n)*(1-x^n-x^(2*n))).

More terms from Emeric Deutsch, Sep 08 2007

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cp's OEIS Frontend

A168019 Square array A(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k+1.

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A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

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A137587 Triangle read by rows: A051731 * A026794.

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A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

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A130708 Number of compositions of n such that every part divides the largest part.

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A130711 Number of compositions of n such that the smallest part divides every part.

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