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

Author

Reinhard Zumkeller, Aug 09 2004

Keywords

• nonn
• tabl

Comments

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

Examples

			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).
		

Links

• 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.

Crossrefs

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

Programs

Haskell

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

Maple

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

Mathematica

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 *)

Formula

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