A116685 -id:A116685 - cp's OEIS frontend

A097364 Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k.

1, 2, 0, 2, 1, 0, 3, 1, 1, 0, 2, 3, 1, 1, 0, 4, 2, 3, 1, 1, 0, 2, 5, 3, 3, 1, 1, 0, 4, 4, 6, 3, 3, 1, 1, 0, 3, 6, 6, 7, 3, 3, 1, 1, 0, 4, 6, 10, 7, 7, 3, 3, 1, 1, 0, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 0, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 0, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 0, 4, 10, 21, 22, 24, 17

Offset: 1

Reinhard Zumkeller, Aug 09 2004

Sum_{k=0..n-1} T(n,k) = A000041(n); T(n,0) + T(n,1) = n for n > 1;
T(n,0) = A000005(n); T(n,1) = A049820(n) for n > 1;
T(n,2) = floor((n-2)/2)*(floor((n-2)/2) + 1)/2 = A000217(floor((n-2)/2)) = A008805(n-4) for n > 3.
Without the 0's (which are of no consequence for the triangle) this sequence is A116685. - Emeric Deutsch, Feb 23 2006

			Triangle starts:
01:  1
02:  2  0
03:  2  1  0
04:  3  1  1  0
05:  2  3  1  1  0
06:  4  2  3  1  1  0
07:  2  5  3  3  1  1 0
08:  4  4  6  3  3  1 1 0
09:  3  6  6  7  3  3 1 1 0
10:  4  6 10  7  7  3 3 1 1 0
11:  2  9 10 12  8  7 3 3 1 1 0
12:  6  6 15 14 13  8 7 3 3 1 1 0
13:  2 11 15 20 16 14 8 7 3 3 1 1 0
14:  4 10 21 22 24 17 ...
- _Joerg Arndt_, Feb 22 2014
T(8,0)=4: 8=4+4=2+2+2+2=1+1+1+1+1+1+1+1,
T(8,1)=4: 3+3+2=2+2+2+1+1=2+2+1+1+1+1=2+1+1+1+1+1+1,
T(8,2)=6: 5+3=4+2+2=3+3+1+1=3+2+2+1=3+2+1+1+1=3+1+1+1+1+1,
T(8,3)=3: 4+3+1=4+2+1+1=4+1+1+1+1,
T(8,4)=3: 6+2=5+2+1=5+1+1+1,
T(8,5)=1: 6+1+1,
T(8,6)=1: 7+1,
T(8,7)=0;
Sum_{k=0..7} T(8,k) = 4+4+6+3+3+1+1+0 = 22 = A000041(8).

Reinhard Zumkeller, Rows n = 1..60 of triangle, flattened
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

Cf. A116685 (same sequence with zeros omitted).

Columns k=3..10 give A128508, A218567, A218568, A218569, A218570, A218571, A218572, A218573. T(2*n,n) = A117989(n). - Alois P. Heinz, Nov 02 2012

a097364 n k = length [qs | qs <- pss !! n, last qs - head qs == k] where
   pss = [] : map parts [1..] where
         parts x = [x] : [i : ps | i <- [1..x],
                                   ps <- pss !! (x - i), i <= head ps]
a097364_row n = map (a097364 n) [0..n-1]
a097364_tabl = map a097364_row [1..]
-- Reinhard Zumkeller, Feb 01 2013

g:=sum(x^i/(1-x^i)/product(1-t*x^j,j=1..i-1),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n],t,j),j=0..n-1) od;
# yields sequence in triangular form # Emeric Deutsch, Feb 23 2006

rows = 14; max = rows+2; col[k0_ /; k0 > 0] := col[k0] = Sum[x^(2*k + k0) / Product[(1-x^(k+j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x]&; col[0] := Table[Switch[n, 1, 0, 2, 1, , n - 1 - col[1][[n]]], {n, 1, Length[col[1]]}]; Table[col[k][[n+2]], {n, 0, rows-1 }, {k, 0, n}] // Flatten (* _Jean-François Alcover, Sep 10 2017, after Alois P. Heinz *)

G.f.: Sum_{i>=1} x^i/((1 - x^i)*Product_{j=1..i-1} (1 - t*x^j)). - Emeric Deutsch, Feb 23 2006

A116686 Total number of parts smaller than the largest part, in all partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 8, 15, 29, 48, 79, 123, 188, 276, 404, 575, 808, 1122, 1540, 2089, 2811, 3748, 4958, 6519, 8504, 11034, 14231, 18268, 23312, 29638, 37486, 47245, 59279, 74140, 92347, 114703, 141933, 175174, 215478, 264407, 323448, 394788, 480509, 583609

Offset: 1

Emeric Deutsch, Feb 23 2006

nonn

Also, sum over all partitions of n of the difference between the largest part and the smallest part. - Franklin T. Adams-Watters, Feb 29 2008

Row sums of A244966. - Omar E. Pol, Jul 19 2014

			a(5) = 8 because the partitions of 5 are [5], [4,(1)], [3,(2)], [3,(1),(1)], [2,2,(1)], [2,(1),(1),(1)] and [1,1,1,1,1], containing a total of 8 parts that are smaller than the largest part (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A116685, A211870, A211881.

f:=sum(x^i*sum(x^j/(1-x^j),j=1..i-1)/product(1-x^q,q=1..i),i=2..55): fser:=series(f,x=0,50): seq(coeff(fser,x^n),n=1..47);

Table[Length[Flatten[Rest[Split[#]]&/@Select[IntegerPartitions[n], #[[1]]> #[[-1]]&]]],{n,50}] (* Harvey P. Dale, Jul 26 2016 *)

a(n) = Sum_{k>=0} k*A116685(n,k).
G.f.: Sum_{i>=1} (x^i*(Sum_{j=1..i-1} x^j/(1-x^j))/(Product_{q=1..i} (1-x^q))).
a(n) = A006128(n) - A046746(n). - Vladeta Jovovic, Feb 24 2006
a(n) = A211870(n) + A211881(n). - Alois P. Heinz, Feb 13 2013

A128508 Number of partitions p of n such that max(p) - min(p) = 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 3, 7, 7, 12, 14, 20, 22, 32, 34, 45, 51, 63, 69, 87, 93, 112, 124, 144, 156, 184, 196, 225, 245, 275, 295, 335, 355, 396, 426, 468, 498, 552, 582, 637, 679, 735, 777, 847, 889, 960, 1016, 1088, 1144, 1232, 1288, 1377, 1449, 1539, 1611, 1719

Offset: 0

John W. Layman, May 07 2007

nonn

See A008805 and A049820 for the numbers of partitions p of n such that max(p)-min(p)=1 or 2, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Cf. A008805, A049820, A097364, A116685, A117142, A117143.

np[n_]:=Length[Select[IntegerPartitions[n],Max[#]-Min[#]==3&]]; Array[np,60] (* Harvey P. Dale, Jul 02 2012 *)

Conjecture. a(1)=0 and, for n>1, a(n+1)=a(n)+d(n), where d(n) is defined as follows: d=0,0,0,1,0 for n=1,...,5 and, for n>5, d(n)=d(n-2)+1 if n=6k or n=6k+4, d(n)=d(n-2) if n=6k+1 or n=6k+3, d(n)=d(n-2)+2Floor[n/6] if n=6k+2 and d(n)=d(n-5) if n=6k+5.
G.f. for number of partitions p of n such that max(p)-min(p) = m is Sum_{k>0} x^(2*k+m)/Product_{i=0..m} (1-x^(k+i)). - Vladeta Jovovic, Jul 04 2007
a(n) = A097364(n,3) = A116685(n,3) = A117143(n) - A117142(n). - Alois P. Heinz, Nov 02 2012

More terms from Vladeta Jovovic, Jul 04 2007

A218567 Number of partitions p of n such that max(p)-min(p) = 4.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 13, 16, 24, 27, 40, 46, 60, 71, 92, 103, 131, 149, 181, 206, 247, 275, 329, 366, 424, 474, 548, 601, 690, 759, 858, 942, 1059, 1152, 1293, 1404, 1555, 1690, 1869, 2013, 2218, 2390, 2614, 2812, 3066, 3282, 3574, 3820, 4131, 4415, 4769, 5071

Offset: 6

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 52; offset = 6; max = terms + offset; s[k0_ /; k0>0] := Sum[x^(2*k + k0) / Product[ (1 - x^(k+j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x]&; Drop[s[4], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]==4&)],{n,6,60}] (* _Harvey P. Dale, Jul 10 2021 *)

G.f.: Sum_{k>0} x^(2*k+4)/Product_{j=0..4} (1-x^(k+j)).

a(n) = A097364(n,4) = A116685(n,4) = A194621(n,4) - A194621(n,3) = A218506(n) - A117143(n).

A218573 Number of partitions p of n such that max(p) - min(p) = 10.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 67, 93, 119, 161, 201, 267, 332, 428, 531, 674, 824, 1034, 1258, 1552, 1877, 2294, 2749, 3332, 3970, 4762, 5645, 6723, 7916, 9367, 10974, 12894, 15036, 17571, 20381, 23696, 27370, 31652, 36416, 41926, 48029, 55071, 62860

Offset: 12

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 48; offset = 12; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[10], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]==10&)],{n,12,60}] (* _Harvey P. Dale, Aug 14 2024 *)

G.f.: Sum_{k>0} x^(2*k+10)/Product_{j=0..10} (1-x^(k+j)).

a(n) = A097364(n,10) = A116685(n,10) = A194621(n,10) - A194621(n,9) = A218512(n) - A218511(n).

A264402 Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 4, 1, 4, 5, 2, 2, 8, 4, 1, 4, 9, 7, 2, 3, 12, 10, 4, 1, 4, 14, 15, 7, 2, 2, 17, 20, 12, 4, 1, 6, 18, 27, 17, 7, 2, 2, 23, 33, 26, 12, 4, 1, 4, 24, 44, 35, 19, 7, 2, 4, 27, 51, 49, 28, 12, 4, 1, 5, 30, 64, 63, 41, 19, 7, 2, 2

Offset: 1

Emeric Deutsch, Nov 21 2015

T(n,k) = number of partitions of n in which the 2nd largest part is k (0 if all parts are equal). Example: T(7,2) = 4 because we have [3,2,1,1], [3,2,2], [4,2,1], and [5,2].
The fact that the above two statistics (in Name and in 1st Comment) have the same distribution follows at once by conjugation. - Emeric Deutsch, Dec 11 2015
Row sums yield the partition numbers (A000041).
T(n,0) = A000005(n) = number of divisors of n.
Sum_{k>=0} k*T(n,k) = A182984(n).

			T(7,2) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,2,1].
Triangle starts:
1;
2;
2,1;
3,2;
2,4,1;
4,5,2;
2,8,4,1;

Alois P. Heinz, Rows n = 1..350, flattened

Cf. A000005, A000041, A116685, A182984, A002541.

g := sum(x^i/((1-x^i)*(product(1-t*x^j, j = i+1 .. 100))), i = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n to 27 do P[n] := sort(coeff(gser, x, n)) end do: for n to 27 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, b(n, i-1) +add((p->[0, p[1]+
       expand(p[2]*x^j)])(b(n-i*j, i-1)) , j=1..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):
seq(T(n), n=1..20);  # Alois P. Heinz, Nov 29 2015

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Sum[ Function[p, {0, p[[1]] + Expand[p[[2]]*x^j]}][b[n-i*j, i-1]], {j, 1, n/i} ]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n][[2]]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

G.f.: G(t,x) = Sum_{i>=1} (x^i/((1 - x^i)*Product_{j>=i+1}(1-t*x^j))).

A218568 Number of partitions p of n such that max(p)-min(p) = 5.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 17, 26, 31, 45, 54, 73, 87, 114, 135, 171, 200, 249, 290, 352, 406, 487, 560, 660, 752, 879, 997, 1153, 1298, 1489, 1671, 1900, 2121, 2397, 2665, 2992, 3311, 3701, 4081, 4535, 4982, 5514, 6042, 6655, 7265, 7977, 8686, 9502, 10314, 11248

Offset: 7

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Table[Count[IntegerPartitions[n],?(Max[#]-Min[#]==5&)],{n,7,60}] (* _Harvey P. Dale, Sep 25 2015 *)

G.f.: Sum_{k>0} x^(2*k+5)/Product_{j=0..5} (1-x^(k+j)).

a(n) = A097364(n,5) = A116685(n,5) = A194621(n,5) - A194621(n,4) = A218507(n) - A218506(n).

A218569 Number of partitions p of n such that max(p)-min(p) = 6.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 27, 33, 49, 59, 81, 100, 131, 158, 205, 243, 306, 365, 448, 527, 642, 748, 896, 1042, 1231, 1418, 1667, 1906, 2215, 2527, 2909, 3298, 3781, 4260, 4847, 5446, 6158, 6886, 7756, 8633, 9669, 10738, 11970, 13239, 14713, 16212, 17943

Offset: 8

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Table[Count[IntegerPartitions[n],?(First[#]-Last[#]==6&)],{n,8,60}] (* _Harvey P. Dale, Feb 09 2015 *)

G.f.: Sum_{k>0} x^(2*k+6)/Product_{j=0..6} (1-x^(k+j)).

a(n) = A097364(n,6) = A116685(n,6) = A194621(n,6) - A194621(n,5) = A218508(n) - A218507(n).

A218570 Number of partitions p of n such that max(p)-min(p) = 7.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 34, 51, 63, 86, 108, 144, 175, 229, 278, 351, 425, 529, 630, 775, 919, 1109, 1309, 1565, 1827, 2167, 2518, 2952, 3414, 3975, 4563, 5281, 6036, 6931, 7889, 9012, 10200, 11598, 13078, 14785, 16613, 18704, 20925, 23470, 26174, 29229

Offset: 9

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 49; offset = 9; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[7], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: Sum_{k>0} x^(2*k+7)/Product_{j=0..7} (1-x^(k+j)).

a(n) = A097364(n,7) = A116685(n,7) = A194621(n,7) - A194621(n,6) = A218509(n) - A218508(n).

A218571 Number of partitions p of n such that max(p)-min(p) = 8.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 52, 65, 90, 113, 152, 188, 246, 302, 387, 471, 591, 714, 884, 1059, 1292, 1538, 1857, 2193, 2621, 3077, 3646, 4254, 4999, 5801, 6772, 7815, 9062, 10409, 12002, 13719, 15733, 17909, 20438, 23169, 26318, 29722, 33623, 37833

Offset: 10

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 48; offset = 10; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[8], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: Sum_{k>0} x^(2*k+8)/Product_{j=0..8} (1-x^(k+j)).

a(n) = A097364(n,8) = A116685(n,8) = A194621(n,8) - A194621(n,7) = A218510(n) - A218509(n).

cp's OEIS Frontend

A097364 Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k.

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A116686 Total number of parts smaller than the largest part, in all partitions of n.

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A128508 Number of partitions p of n such that max(p) - min(p) = 3.

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A218567 Number of partitions p of n such that max(p)-min(p) = 4.

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A218573 Number of partitions p of n such that max(p) - min(p) = 10.

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A264402 Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

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A218568 Number of partitions p of n such that max(p)-min(p) = 5.

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A218569 Number of partitions p of n such that max(p)-min(p) = 6.

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