A092265 -id:A092265 - cp's OEIS frontend

A034296 Number of flat partitions of n: partitions {a_i} with each |a_i - a_{i-1}| <= 1.

1, 1, 2, 3, 4, 5, 7, 8, 10, 13, 15, 18, 23, 26, 31, 39, 44, 52, 63, 72, 85, 101, 115, 134, 158, 181, 208, 243, 277, 318, 369, 418, 478, 549, 622, 710, 809, 914, 1036, 1177, 1328, 1498, 1695, 1904, 2143, 2416, 2706, 3036, 3408, 3811, 4264, 4769, 5319, 5934, 6621

Offset: 0

Erich Friedman

nonn

Also number of partitions of n such that all parts, with the possible exception of the largest, appear only once. Example: a(6)=7 because we have [6], [5,1], [4,2], [3,3], [3,2,1], [2,2,2] and [1,1,1,1,1,1] ([4,1,1], [3,1,1,1], [2,2,1,1], [2,1,1,1,1,1] do not qualify). - Emeric Deutsch and Vladeta Jovovic, Feb 23 2006
Also the number of partitions p of n such that d(p) > max(p) - min(p), where d is the number of distinct parts of p; indeed that inequality occurs only when d(p) = 1+ max(p) - min(p), so p satisfies a(i) - a(i-1) = 1 for all parts, ordered as a(i) >= a(i-1) > ... > a(k). - Clark Kimberling, Apr 18 2014
Flat partitions are also called gap-free partitions. See, for example, the Grabner et al. reference. - Emeric Deutsch, Sep 22 2016
Conjecture: Also the sum of the smallest parts in the distinct partitions of n with an odd number of parts. - George Beck, May 06 2017
Above conjecture was proved by Shane Chern, see link. - George Beck, Aug 12 2017
Note that Andrews [2016] uses a(0) = 1. - Michael Somos, Aug 07 2017
Also called number of compact partitions of n where a compact partition is one where every integer between the largest and smallest parts of it also appears as a part. [Andrews 2016] - Michael Somos, Aug 13 2017

			From _Joerg Arndt_, Dec 27 2012: (Start)
The a(11)=18 flat partitions of 11 are (in lexicographic order)
[ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 ]
[ 2]  [ 2 1 1 1 1 1 1 1 1 1 ]
[ 3]  [ 2 2 1 1 1 1 1 1 1 ]
[ 4]  [ 2 2 2 1 1 1 1 1 ]
[ 5]  [ 2 2 2 2 1 1 1 ]
[ 6]  [ 2 2 2 2 2 1 ]
[ 7]  [ 3 2 1 1 1 1 1 1 ]
[ 8]  [ 3 2 2 1 1 1 1 ]
[ 9]  [ 3 2 2 2 1 1 ]
[10]  [ 3 2 2 2 2 ]
[11]  [ 3 3 2 1 1 1 ]
[12]  [ 3 3 2 2 1 ]
[13]  [ 3 3 3 2 ]
[14]  [ 4 3 2 1 1 ]
[15]  [ 4 3 2 2 ]
[16]  [ 4 4 3 ]
[17]  [ 6 5 ]
[18]  [ 11 ]
The a(11)=18 partitions of 11 where no part (except possibly the largest) is repeated are
[ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 ]
[ 2]  [ 2 2 2 2 2 1 ]
[ 3]  [ 3 3 3 2 ]
[ 4]  [ 4 4 2 1 ]
[ 5]  [ 4 4 3 ]
[ 6]  [ 5 3 2 1 ]
[ 7]  [ 5 4 2 ]
[ 8]  [ 5 5 1 ]
[ 9]  [ 6 3 2 ]
[10]  [ 6 4 1 ]
[11]  [ 6 5 ]
[12]  [ 7 3 1 ]
[13]  [ 7 4 ]
[14]  [ 8 2 1 ]
[15]  [ 8 3 ]
[16]  [ 9 2 ]
[17]  [ 10 1 ]
[18]  [ 11 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016; See page 4 equation (2.1).
Shane Chern, On a conjecture of George Beck, arXiv:1705.10700 [math.NT], 2017.
P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018. Also Discrete Masth., 343 (2020), # 111875.
Jane Y. X. Yang, Combinatorial proofs and generalizations on conjectures related with Euler's partition theorem, arXiv:1801.06815 [math.CO], 2018.

Cf. A034297, A239954, A092265.

Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), this sequence (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).

g:= 1+sum(x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..50); # Emeric Deutsch, Feb 23 2006
# second Maple program:
b:= 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(b(n, k), k=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Jul 06 2012

nn=54;Drop[CoefficientList[Series[Sum[x^i/(1-x^i)Product[1+x^j,{j,1,i-1}],{i,1,nn}],{x,0,nn}],x],1] (* Geoffrey Critzer, Sep 28 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[b[n, k], {k, 1, n}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ Sum[ x^k / (1 - x^k) QPochhammer[ -x, x, k - 1] // FunctionExpand, {k, n}], {x, 0, n}]; (* Michael Somos, Aug 07 2017 *)

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, x^n/(1-x^n) * prod(k=1,n-1,1+x^k) );
v = Vec(gf)
/* Joerg Arndt, Apr 21 2013 */

{a(n) = my(t); if( n<1, 0, polcoeff(sum(k=1, n, (t *= 1 + x^k) * x^k / (1 - x^(2*k)), t = 1 + x * O(x^n)), n))}; /* Michael Somos, Aug 07 2017 */

{a(n) = my(c); forpart(p=n, c++; for(i=1, #p-1, if( p[i+1] > p[i] + 1, c--; break))); c}; /* Michael Somos, Aug 13 2017 */

from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==0 else 0 if i<1 else sum(b(n - i*j, i - 1) for j in range(1, n//i + 1))
def a(n): return sum(b(n, k) for k in range(n + 1))
print([a(n) for n in range(71)]) # Indranil Ghosh, Aug 14 2017, after Maple code by Alois P. Heinz

G.f.: x/(1-x) + x^2/(1-x^2)*(1+x) + x^3/(1-x^3)*(1+x)*(1+x^2) + x^4/(1-x^4)*(1+x)*(1+x^2)*(1+x^3) + x^5/(1-x^5)*(1+x)*(1+x^2)*(1+x^3)*(1+x^4) + ... . - Emeric Deutsch and Vladeta Jovovic, Feb 22 2006
a(n) = Sum_{k=0..1} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018

More terms from Emeric Deutsch, Feb 23 2006

a(0)=1 prepended by Alois P. Heinz, Aug 14 2017

A336902 Sum of the smallest parts of all compositions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 5, 6, 11, 18, 25, 32, 53, 84, 107, 156, 205, 302, 497, 618, 863, 1206, 1597, 2228, 3569, 4440, 6191, 8256, 11329, 14642, 20477, 30390, 38555, 52578, 69625, 92696, 122141, 160500, 211955, 310476, 386941, 521102, 678617, 901386, 1155383, 1529742, 1940749

Offset: 0

Alois P. Heinz, Aug 07 2020

nonn

			a(6) = 18 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 1 + 1 + 6: (1)23, (1)32, 2(1)3, 23(1), 3(1)2, 32(1), (2)4, 4(2), (1)5, 5(1), (6).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336903.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..50);

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n || i < 1, 0,
     If[i == n, i*p!, b[n-i, Min[n-i, i-1], p+1]] + b[n, i-1, p]];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) == n (mod 2).

A336903 Sum of the largest parts of all compositions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 7, 10, 19, 42, 61, 98, 151, 304, 403, 654, 925, 1400, 2431, 3328, 4903, 7056, 10117, 13952, 23419, 30406, 44683, 61308, 87289, 116822, 164359, 247774, 327715, 457542, 624445, 855062, 1148023, 1559188, 2058643, 3043506, 3906637, 5375732, 7111975, 9679852

Offset: 0

Alois P. Heinz, Aug 07 2020

nonn

			a(6) = 42 = 3 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 6: 12(3), 1(3)2, 21(3), 2(3)1, (3)12, (3)21, 2(4), (4)2, 1(5), (5)1, (6).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336902.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2, 0)):
seq(a(n), n=0..50);

b[n_, i_, p_] := b[n, i, p] = If[i(i + 1)/2 < n, 0,
     If[n == 0, p!, b[n - i, Min[n - i, i - 1], p + 1]*
     If[p == 0, i, 1] + b[n, i - 1, p]]];
a[n_] := If[n == 0, 0, b[n, n, 0]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) == n (mod 2).

A026832 Number of partitions of n into distinct parts, the least being odd.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372, 3766, 4202, 4682

Offset: 0

Clark Kimberling

Fine's numbers L(n).

Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1], [2,2,2,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 29 2006

			a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.

Cf. A000009, A096765.

a026832 n = p 1 n where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012

g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]]  (* Jean-François Alcover, Apr 05 2011, after Emeric Deutsch *)
Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* Harvey P. Dale, May 14 2022 *)

G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic, Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - Emeric Deutsch, Mar 29 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 29 2006

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019

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

A026821 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the least being k, for k=1,2,...,n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 1, 3, 1, 1, 0, 0, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 0, 1, 5, 2, 1, 1, 0, 0, 0, 0, 0, 1, 5, 3, 1, 1, 1, 0, 0, 0, 0, 0, 1, 7, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 8, 4, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling

T(n,1) = A025147(n-1). Sum(k*T(n,k),k=1..n) = A092265(n). - Emeric Deutsch, Feb 24 2006

			T(11,2)=3 because we have [9,2],[6,3,2] and [5,4,2].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,1;
1,1,0,0,1;

Cf. A025147, A092265.

g:=sum(t^j*x^j*product(1+x^i,i=j+1..50),j=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: seq(seq(coeff(P[n],t^j),j=1..n),n=1..14); # Emeric Deutsch, Feb 24 2006

T(n, k) = T(n-k, k+1) + ... + T(n-k, n-k) for 1<=k<=m and T(n, k)=0 for m+1<=k<=n-1, where m=[ (n-1)/2 ]; T(n, n)=1 for n >= 1.

G.f.: sum(t^j*x^j*product(1+x^i,i=j+1..infinity),j=1..infinity). - Emeric Deutsch, Feb 24 2006

A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 12, 19, 27, 41, 54, 76, 99, 133, 171, 223, 279, 357, 443, 554, 682, 841, 1022, 1247, 1504, 1814, 2174, 2603, 3092, 3676, 4346, 5127, 6030, 7076, 8275, 9669, 11254, 13078, 15167, 17556, 20270, 23377, 26899, 30902, 35448, 40592, 46403

Offset: 1

Emeric Deutsch, Mar 18 2006

nonn

a(n) = sum(k*A117454(n,k), k=0..n-2).

a(n) = A005895(n)-A092265(n). - Alois P. Heinz, Jul 06 2012

			a(7)=12 because the partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] and (7-7)+(6-1)+(5-2)+(4-3)+(4-1)=12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A005895, A092265, A117454.

g:=sum(x^(i*(i+1)/2)*sum(1/(1-x^j),j=1..i-1)/product(1-x^j,j=1..i),i=1..15): gser:=series(g,x=0,55): seq(coeff(gser,x^n), n=1..50);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(i=n, n, 0)+`if`(i>0, b(n, i-1)+
      `if`(i g(n, 1) -b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 06 2012

b[n_, i_] := b[n, i] = If[i==n, n, 0] + If[i>0, b[n, i-1] + If[iJean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

G.f.: sum(x^(i(i+1)/2)*sum(1/(1-x^j), j=1..i-1)/product(1-x^j, j=1..i), i=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117454 and letting t=1).

A277230 Irregular triangular array T(n, k) giving in row n the base of the Ferrers diagram of the k-th partition of n into distinct parts. The partitions of n are taken in Abramowitz-Stegun order but with decreasing parts. T(n, k) is the smallest part of the k-th partition of n into distinct parts.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 1, 2, 6, 1, 2, 1, 7, 1, 2, 3, 1, 8, 1, 2, 3, 1, 1, 9, 1, 2, 3, 4, 1, 1, 2, 10, 1, 2, 3, 4, 1, 1, 1, 2, 1, 11, 1, 2, 3, 4, 5, 1, 1, 1, 2, 2, 1, 12, 1, 2, 3, 4, 5, 1, 1, 1, 1, 2, 2, 3, 1, 1, 13, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 14, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 1, 1, 1, 1, 2, 15, 1, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Wolfdieter Lang, Oct 21 2016

The length of row n of this irregular triangular array is A000009(n).
For the Abramowitz-Stegun order of partitions see an Apr 04 2011 comment on A036036.
The sum of the numbers of row n is A092265(n).
See the Hardy (H) and Hardy-Wright (H-W) references, where the base is called beta. The companion array is A277231 giving the slopes (called sigma) of these partitions with distinct parts. These beta and sigma numbers play a role in an elementary proof of Euler's pentagonal-number theorem (pp. 284-287 in (H-W), and pp. 83-85 in (H)) by F. Franklin from 1881. See also MacMahon and Charalambides.
The base of the Ferrers diagram of the k-th partition of n into distinct parts (in the mentioned order) is the number of nodes in the last row, the smallest part of the partition.
The slope of such a partition is the number of nodes on the NE-SW diagonal through the last node of the first row of the Ferrers diagram. (The name may be misleading. The usual slope of the NE-SW diagonal is of course 1).
The number of parts m of these partitions is from m = 1, 2, ..., A003056(n).

			The irregular triangle begins (brackets separate partitions with equal number of parts m = 1, 2, 3,..., A003056(n)):
n\k   1   2  3   4   5   6   7  8  9  10 ...
1:   [1]
2:   [2]
3:   [3] [1]
4:   [4] [1]
5:   [5] [1, 2]
6:   [6] [1, 2] [1]
7:   [7] [1, 2,  3] [1]
8:   [8] [1, 2,  3] [1,  1]
9:   [9] [1, 2,  3,  4] [1,  1, 2]
10: [10] [1, 2,  3,  4] [1,  1, 1, 2] [1]
...
n = 11: [11] [1, 2,  3,  4, 5] [1, 1, 1, 2, 2] [1],
n = 12: [12] [1, 2, 3, 4, 5] [1, 1, 1, 1, 2, 2, 3] [1, 1],
n = 13: [13] [1, 2, 3, 4, 5, 6] [1, 1, 1, 1, 2, 2, 2, 3] [1, 1, 1],
n = 14: [14] [1, 2, 3, 4, 5, 6] [1, 1, 1, 1, 1, 2, 2, 2, 3, 3] [1, 1, 1, 1, 2],
n = 15: [15] [1, 2, 3, 4, 5, 6, 7] [1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4] [1, 1, 1, 1, 1, 2] [1].
----------------------------------------
The partition of n = 5 + 4 + 1 = 10 has base 1 and slope 2 (beta < sigma):
o o o o o
o o o o
o
The partition of n = 5 + 3 + 1 = 9 has base 1 and slope 1 (beta = sigma):
o o o o o
o o o
o
The partition of n = 5 + 3 + 2 = 10 has base 2 and slope 1 (beta > sigma):
o o o o o
o o o
o o
------------------------------------------
The partitions of n = 6 with m = 1, 2, and 3,  (3 = A003056(6)) distinct parts are:
[6], [[5, 1], [4, 2]], [3, 2, 1], with base numbers in row n=6: [6] [1, 2] [1]
and slope numbers in row n=6 of A277231:
[1] [1, 1] [3].

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, pp. 389-391, 396, 595.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 83-85.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 284-287.
P. A. MacMahon, Combinatory Analysis, Vol. II, Chelsea Publishing Company, New York, 1960, pp. 21-23.

F. Franklin, Sur le développement du produit infini (1-x) (1-x^2) (1-x^3) ..., Comptes Rendus de l'Académie des Sciences, Paris, 92 (1881) 448-450.

Cf. A000009, A003056, A092265, A277231.

Table[Function[w, Flatten@ Map[Function[k, Min /@ Select[w, Length@ # == k &]], Range@ Max@ Map[Length, w]]]@ Select[ DeleteCases[ IntegerPartitions@ n, w_ /; MemberQ[Differences@ w, 0]], Length@ # <= Floor[(Sqrt[1 + 8 n] - 1)/2] &], {n, 15}] // Flatten (* Michael De Vlieger, Oct 26 2016 *)

T(n, k) is the smallest part of the k-th partition of n into distinct parts. n >=1. k=1, 2, ..., A000009(n). Partitions appear in Abramowitz-Stegun order.

cp's OEIS Frontend

A034296 Number of flat partitions of n: partitions {a_i} with each |a_i - a_{i-1}| <= 1.

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A336902 Sum of the smallest parts of all compositions of n into distinct parts.

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A336903 Sum of the largest parts of all compositions of n into distinct parts.

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A026832 Number of partitions of n into distinct parts, the least being odd.

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

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A026821 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the least being k, for k=1,2,...,n.

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A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.

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