A135298 a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers.

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 0, 1, 3, 1, 4, 2, 2, 2, 4, 1, 3, 1, 0, 0, 0, 0, 1, 4, 3, 6, 7, 6, 4, 10, 6, 10, 6, 10, 6, 10, 4, 6, 7, 6, 3, 4, 1, 1, 5, 6, 9, 16, 12, 14, 24, 20, 21, 23, 28, 24, 34, 20, 32, 42, 29, 29, 42, 32, 20, 34, 24, 28, 23, 21, 20, 25, 20, 22, 30, 38

Offset: 0

Author

Leroy Quet, Dec 04 2007

Keywords

• nonn
• look

Comments

Does every integer greater than some positive integer N have at least one such representation?

a(n) > 0 for n > 34, a(n) > 1 for n > 56. - Alois P. Heinz, Aug 28 2014

Examples

			21 has a(21)=3 such representations: 21 = 1*4 + 2*3 + 3*1 + 4*2 = 1*4 + 2*2 + 3*3 + 4*1 = 1*3 + 2*4 + 3*2 + 4*1.
Not all representations of an integer n need to necessarily have the same j. For example, 91 = 1*1 + 2*2 + 3*3 + 4*4 + 5*5 + 6*6 (j=6). And 91 also equals 1*7 + 2*4 + 3*5 + 4*3 + 5*6 + 6*2 + 7*1 (j=7).
1 = 1*1;
4 = 1*2+2*1;
5 = 1*1+2*2;
10 = 1*3+2*2+3*1;
11 = 1*2+2*3+3*1;
11 = 1*3+2*1+3*2;
13 = 1*1+2*3+3*2;
13 = 1*2+2*1+3*3;
14 = 1*1+2*2+3*3;
20 = 1*4+2*3+3*2+4*1;
21 = 1*3+2*4+3*2+4*1;
21 = 1*4+2*2+3*3+4*1;
21 = 1*4+2*3+3*1+4*2;
22 = 1*3+2*4+3*1+4*2;
23 = 1*2+2*4+3*3+4*1;
23 = 1*3+2*2+3*4+4*1;
23 = 1*4+2*1+3*3+4*2;
23 = 1*4+2*2+3*1+4*3;
24 = 1*2+2*3+3*4+4*1;
24 = 1*4+2*1+3*2+4*3;
25 = 1*2+2*4+3*1+4*3;
25 = 1*3+2*1+3*4+4*2;
26 = 1*1+2*4+3*3+4*2;
26 = 1*3+2*2+3*1+4*4;
27 = 1*1+2*3+3*4+4*2;
27 = 1*1+2*4+3*2+4*3;
27 = 1*2+2*3+3*1+4*4;
27 = 1*3+2*1+3*2+4*4;
28 = 1*2+2*1+3*4+4*3;
29 = 1*1+2*2+3*4+4*3;
29 = 1*1+2*3+3*2+4*4;
29 = 1*2+2*1+3*3+4*4;
30 = 1*1+2*2+3*3+4*4;
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1770
• FindStat - Combinatorial Statistic Finder, The rank of the permutation inside the alternating sign matrix lattice
• J. Sack and H. Úlfarsson, Refined inversion statistics on permutations, arXiv preprint arXiv:1106.1995 [math.CO], 2011-2012.

Crossrefs

Cf. A000292, A126972, A175929.

Programs

Maple

A135298rec := proc(j,n,notm) local a,m ; a := 0 ; if n = 0 then if max( seq(e,e=notm) ) >= j then RETURN(0) ; else RETURN(1) ; fi ; end: for m from 1 do if n-j*m < 0 then break ; elif not m in notm then a := a+A135298rec(j+1,n-j*m,[op(notm),m] ) ; fi ; od: RETURN(a) ; end: A135298 := proc(n) A135298rec(1,n,[]) ; end: for n from 1 to 140 do printf("%d, ",A135298(n)) ; od: # R. J. Mathar, Jan 30 2008
# second Maple program:
n:= 8 : # gives binomial(n+3, 3) terms
with(combinat):
(p-> seq(coeff(p, x, j), j=0..binomial(n+3, 3)-1))
(add(add(x^add(i*l[i], i=1..h), l=permute(h)), h=0..n));
# Alois P. Heinz, Aug 29 2014

Mathematica

n = 8; (* gives binomial(n+3, 3)-1 terms *) Function[p, Table[ Coefficient[p, x, j], {j, 1, Binomial[n+3, 3]-1}]] @ Sum[x^(l.Range[h]), {h, 1, n}, {l, Permutations @ Range[h]}] (* Jean-François Alcover, Jul 22 2017, after Alois P. Heinz *)

Extensions

More terms from R. J. Mathar, Jan 30 2008

a(0)=1 prepended by Alois P. Heinz, Nov 23 2023