A238353 -id:A238353 - cp's OEIS frontend

A238867 Number of partitions of n where the difference between consecutive parts is at most 7.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 95, 125, 162, 210, 268, 344, 434, 549, 688, 861, 1069, 1328, 1637, 2016, 2472, 3023, 3682, 4479, 5424, 6558, 7905, 9508, 11404, 13657, 16307, 19440, 23123, 27454, 32526, 38479, 45424, 53545, 63006, 74024, 86824, 101701, 118931, 138899, 161983, 188656, 219419, 254895, 295709

Offset: 0

Joerg Arndt, Mar 08 2014

nonn

Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most seven times (by taking conjugates).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), A238866 (d=6), this sequence, A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=0..min(7, n/i))))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> add(g(n, k), k=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[7, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

N=66;  q = 'q + O('q^N);
Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, (1-q^(8*i))/(1-q^i) ) ) )

G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(8*i))/(1-q^i) ) ).
a(n) = Sum_{k=0..7} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ 7^(1/4) * exp(Pi*sqrt(7*n/12)) / (2^(7/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

A238868 Number of partitions of n where the difference between consecutive parts is at most 8.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 97, 129, 166, 217, 276, 356, 449, 572, 715, 900, 1117, 1393, 1717, 2123, 2601, 3193, 3889, 4744, 5748, 6970, 8404, 10135, 12165, 14600, 17448, 20845, 24813, 29522, 35009, 41491, 49031, 57900, 68195, 80258, 94234, 110553, 129421, 151382, 176724, 206132, 240002, 279195, 324255

Offset: 0

Joerg Arndt, Mar 08 2014

nonn

Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most eight times (by taking conjugates).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), this sequence, A238869 (d=9), A000041 (d --> infinity).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=0..min(8, n/i))))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> add(g(n, k), k=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[8, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

N=66;  q = 'q + O('q^N);
Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, (1-q^(9*i))/(1-q^i) ) ) )

G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(9*i))/(1-q^i) ) ).
a(n) = Sum_{k=0..8} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ exp(4*Pi*sqrt(n/27)) / (sqrt(2) * 3^(7/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

A238869 Number of partitions of n where the difference between consecutive parts is at most 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 131, 170, 221, 283, 364, 461, 586, 737, 926, 1154, 1439, 1779, 2199, 2703, 3317, 4051, 4942, 6001, 7278, 8796, 10610, 12760, 15323, 18344, 21928, 26148, 31127, 36971, 43848, 51890, 61321, 72327, 85183, 100149, 117588, 137827, 161343, 188583, 220139, 256607, 298761, 347360

Offset: 0

Joerg Arndt, Mar 08 2014

nonn

Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most nine times (by taking conjugates).

In general, for d > 0, "number of partitions with max diff d" is asymptotic to d^(1/4) * exp(Pi*sqrt(2*d*n/(3*(d+1)))) / (2^(5/4) * 3^(1/4) * (d+1)^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), this sequence, A000041 (d --> infinity).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=0..min(9, n/i))))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> add(g(n, k), k=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[9, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(Max[Abs[Differences[#]]]<10&)],{n,0,60}] (* _Harvey P. Dale, Nov 24 2024 *)

N=66;  q = 'q + O('q^N);
Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, (1-q^(10*i))/(1-q^i) ) ) )

G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(10*i))/(1-q^i) ) ).
a(n) = Sum_{k=0..9} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ 3^(1/4) * exp(Pi*sqrt(3*n/5)) / (4 * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022

A238354 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with minimal ascent k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 1, 0, 0, 5, 1, 0, 1, 0, 0, 8, 1, 1, 0, 1, 0, 0, 11, 2, 0, 1, 0, 1, 0, 0, 17, 2, 1, 0, 1, 0, 1, 0, 0, 23, 3, 1, 1, 0, 1, 0, 1, 0, 0, 33, 4, 2, 0, 1, 0, 1, 0, 1, 0, 0, 45, 5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 63, 6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 84, 8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Column k=0: T(n,0) = 1 + A047967(n).
Column k=1 is A238708.
Row sums are A000041.

			Triangle starts:
  00:    1;
  01:    1,  0;
  02:    2,  0, 0;
  03:    2,  1, 0, 0;
  04:    4,  0, 1, 0, 0;
  05:    5,  1, 0, 1, 0, 0;
  06:    8,  1, 1, 0, 1, 0, 0;
  07:   11,  2, 0, 1, 0, 1, 0, 0;
  08:   17,  2, 1, 0, 1, 0, 1, 0, 0;
  09:   23,  3, 1, 1, 0, 1, 0, 1, 0, 0;
  10:   33,  4, 2, 0, 1, 0, 1, 0, 1, 0, 0;
  11:   45,  5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  12:   63,  6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  13:   84,  8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  14:  114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  15:  150, 13, 4, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  ...
The 11 partitions of 6 together with their minimal ascents are:
  01:  [ 1 1 1 1 1 1 ]   0
  02:  [ 1 1 1 1 2 ]     0
  03:  [ 1 1 1 3 ]       0
  04:  [ 1 1 2 2 ]       0
  05:  [ 1 1 4 ]         0
  06:  [ 1 2 3 ]         1
  07:  [ 1 5 ]           4
  08:  [ 2 2 2 ]         0
  09:  [ 2 4 ]           2
  10:  [ 3 3 ]           0
  11:  [ 6 ]             0
There are 8 partitions of 6 with min ascent 0, 1 with min ascents 1, 2, and 4, giving row 6 of the triangle: 8, 1, 1, 0, 1, 0, 0.

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

Cf. A238353 (partitions by maximal ascent).

b:= proc(n, i, t) option remember; `if`(n=0, 1/x, `if`(i<1, 0,
      b(n, i-1, t)+`if`(i>n, 0, (p->`if`(t=0, p, add(coeff(
       p, x, j)*x^`if`(j<0, t-i, min(j, t-i)),
       j=-1..degree(p))))(b(n-i, i, i)))))
    end:
T:= n->(p->seq(coeff(p, x, k)+`if`(k=0, 1, 0), k=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1/x, If[i<1, 0, b[n, i-1, t]+If[i>n, 0, Function[{p}, If[t == 0, p, Sum[Coefficient[p, x, j]*x^If[j<0, t-i, Min[j, t-i]], {j, -1, Exponent[p, x]}]]][b[n-i, i, i]]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, k]+If[k == 0, 1, 0], {k, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)

A251729 Number of gap-free but not complete compositions of n.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 6, 6, 14, 12, 27, 33, 58, 86, 134, 210, 323, 539, 810, 1371, 2044, 3510, 5263, 8927, 13702, 22870, 35821, 58750, 93343, 152236, 243244, 395078, 634342, 1027876, 1656543, 2676693, 4325727, 6982440, 11299457, 18232217, 29518334, 47641410

Offset: 1

Alois P. Heinz, Dec 07 2014

nonn

A composition is gap-free but not complete if all integers in the interval defined by the smallest and the largest part are parts but 1 is not a part.

			a(6) = 3: [6], [3,3], [2,2,2].
a(7) = 6: [7], [3,4], [4,3], [2,2,3], [2,3,2], [3,2,2].

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Alois P. Heinz, Plot of (a(n)+a(n+1)-a(n+2))/a(n+2) for n = 150..1000
P. Hitczenko and A. Knopfmacher, Gap-free compositions and gap-free samples of geometric random variables, Discrete Math., 294 (2005), 225-239.

Cf. A001622, A034296, A107428, A107429, A188575, A238353, A264396.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, 0, t!),
     `if`(i<1 or n add(b(n, i, 0), i=1..n):
seq(a(n), n=1..50);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, 0, t!], If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1, t + j]/j!, {j, 1, n/i}]]];
a[n_] := Sum[b[n, i, 0], {i, 1, n}];
Array[a, 50] (* Jean-François Alcover, Jan 25 2021, after Alois P. Heinz *)

a(n) = A107428(n) - A107429(n).

lim_{n -> oo} a(n)/a(n-1) = (1+sqrt(5))/2 = phi = A001622.

A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 0, 6, 1, 1, 0, 7, 0, 8, 0, 2, 1, 9, 0, 0, 1, 0, 0, 10, 1, 11, 0, 2, 1, 1, 0, 12, 1, 2, 0, 13, 1, 14, 0, 0, 1, 15, 0, 0, 0, 2, 0, 16, 0, 2, 0, 2, 1, 17, 0, 18, 1, 0, 0, 3, 1, 19, 0, 2, 1, 20, 0, 21, 1, 0, 0, 1, 1, 22, 0, 0, 1, 23

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The 0-prepended prime indices of 9842 are {0,1,4,8,12}, with differences (1,3,4,4), so a(9842) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of positive terms are A005117, complement A013929.
A similar statistic is counted by A238353.
The maximal version is A286469, without prepending A355526.
Without prepending we have A355524 or A355525.
Positions of ones are A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A112798 lists prime indices, with sum A056239.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A064428, A066312, A091602, A120944, A238354, A286470, A325161, A352822, A355527, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min@@Differences[Prepend[primeMS[n],0]],{n,2,100}]

A237665 Number of partitions of n such that the distinct terms arranged in increasing order form a string of two or more consecutive integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 17, 24, 27, 35, 39, 50, 57, 70, 79, 97, 111, 132, 150, 178, 204, 239, 271, 316, 361, 416, 472, 545, 618, 706, 800, 912, 1032, 1173, 1320, 1496, 1687, 1902, 2137, 2410, 2702, 3034, 3398, 3808, 4258, 4765, 5313, 5932, 6613

Offset: 0

Clark Kimberling, Feb 11 2014

Number of partitions of n with maximal distance between parts = 1; column k=1 of A238353. [Joerg Arndt, Mar 23 2014]

Conjecture: a(n+1) = sum of smallest parts in the distinct partitions of n with an even number of parts. - George Beck, May 06 2017

			The qualifying partitions of 8 are 332, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 6.  (The strings of distinct parts arranged in increasing order are 23, 123, 123, 12, 12, 12.)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Shane Chern (Xiaohang Chen), On a conjecture of George Beck. II, 2018.

Cf. A034296, A237666, A092265.

b:= proc(n, i, t) option remember;
      `if`(n=0 or i=1, `if`(n=0 and t=2 or n>0 and t>0, 1, 0),
      `if`(i>n, 0, add(b(n-i*j, i-1, min(t+1, 2)), j=1..n/i)))
    end:
a:= n-> add(b(n, i, 0), i=1..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 15 2014

Map[Length[Select[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], (Table[-1, {Length[#]}] == # && # =!= \{}) &]] &, Range[55]] (* Peter J. C. Moses, Feb 09 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0 || i==1, If[n==0 && t==2 || n>0 && t > 0, 1, 0], If[i>n, 0, Sum[b[n-i*j, i-1, Min[t+1, 2]], {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jan 28 2022

A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

Original entry on oeis.org

6, 15, 30, 35, 42, 66, 70, 77, 78, 102, 105, 114, 138, 143, 154, 165, 174, 186, 195, 210, 221, 222, 231, 246, 255, 258, 282, 285, 286, 318, 323, 330, 345, 354, 366, 385, 390, 402, 426, 429, 435, 437, 438, 442, 455, 462, 465, 474, 498, 510, 534, 546, 555, 570

Offset: 1

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
    6: {1,2}
   15: {2,3}
   30: {1,2,3}
   35: {3,4}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   77: {4,5}
   78: {1,2,6}
  102: {1,2,7}
  105: {2,3,4}
  114: {1,2,8}
  138: {1,2,9}
  143: {5,6}
  154: {1,4,5}
  165: {2,3,5}
  174: {1,2,10}
  186: {1,2,11}
  195: {2,3,6}
  210: {1,2,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For minimal difference <= 1 we have A055932.
For maximal instead of minimal difference = 1 we have A066312.
For minimal difference > 1 we have A325160.
If zero is considered a prime index we get A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 or A355525 give minimal difference between prime indices.
Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[primeMS[#]]==1&]

Intersection of A005117 (squarefree) and A104210 (has consecutive primes).

A355523 Number of distinct differences between adjacent prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 0, 2, 0, 1, 2, 2, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 1, 2, 0, 2, 2, 2, 0, 2, 0, 2, 1

Offset: 1

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			For example, the prime indices of 22770 are {1,2,2,3,5,9}, with differences (1,0,1,2,4), so a(22770) = 4.

Crossrefs found in the link are not repeated here.
Counting m such that A056239(m) = n and a(m) = k gives A279945.
With multiplicity we have A252736(n) = A001222(n) - 1.
The maximal difference is A286470, minimal A355524.
A008578 gives the positions of 0's.
A287352 lists differences between 0-prepended prime indices.
A355534 lists augmented differences between prime indices.
A355536 lists differences between prime indices.
Cf. A066312, A238353, A238710, A286469, A320348, A325388, A325406, A351294, A352827, A355525-A355533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Differences[primeMS[n]]]],{n,1000}]

A355523(n) = if(1==n, 0, my(pis = apply(primepi,factor(n)[,1]), difs = vector(#pis-1, i, pis[i+1]-pis[i])); (#Set(difs)+!issquarefree(n))); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A342531 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 2, 3, 1, 1, 1, 1, 0, 1, 0, 0

Offset: 0

Gus Wiseman, Mar 25 2021

The maximal descent of an empty or singleton partition is considered to be 0.

			Triangle begins:
1
1 0
1 0 0
1 1 0 0
1 0 1 0 0
1 1 0 1 0 0
1 1 1 0 1 0 0
1 1 1 1 0 1 0 0
1 0 2 1 1 0 1 0 0
1 2 1 1 1 1 0 1 0 0
1 1 2 2 1 1 1 0 1 0 0
1 1 2 3 1 1 1 1 0 1 0 0
1 1 3 2 3 1 1 1 1 0 1 0 0
1 1 3 3 3 2 1 1 1 1 0 1 0 0
1 1 3 4 3 3 2 1 1 1 1 0 1 0 0
1 3 3 4 4 3 2 2 1 1 1 1 0 1 0 0
1 0 5 5 5 4 3 2 2 1 1 1 1 0 1 0 0
1 1 4 7 5 5 4 2 2 2 1 1 1 1 0 1 0 0
1 2 5 6 7 6 4 4 2 2 2 1 1 1 1 0 1 0 0
1 1 5 9 7 7 6 4 3 2 2 2 1 1 1 1 0 1 0 0
1 1 6 9 9 7 8 5 4 3 2 2 2 1 1 1 1 0 1 0 0
Row n = 15 counts the following strict partitions (empty columns indicated by dots, A..F = 10..15):
  F  87     753   96    762   A5   A41   B4   B31  C3  C21  D2  .  E1  .  .
     654    6432  852   843   861  9321  A32
     54321  6531  7431  951   942
                  7521  8421

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The non-strict version is A238353.
A000041 counts partitions (strict: A000009).
A049980 counts strict partitions with equal differences.
A325325 counts partitions with distinct differences (ranking: A325368).
A325545 counts compositions with distinct differences.
Cf. A003114, A003242, A005117, A034296, A049988, A238710, A342098, A342514.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&If[Length[#]<=1,k==0,Max[Differences[Reverse[#]]]==k]&]],{n,0,15},{k,0,n}]

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