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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..680

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.

EXAMPLE

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;

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)-1 terms

with(combinat):

(p-> seq(coeff(p, x, j), j=1..binomial(n+3, 3)-1))

(add(add(x^add(i*l[i], i=1..h), l=permute(h)), h=1..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 *)

CROSSREFS

Cf. A126972, A175929.

Sequence in context: A137668 A056615 A060989 * A006996 A321430 A343914

Adjacent sequences:  A135295 A135296 A135297 * A135299 A135300 A135301

KEYWORD

nonn,look

AUTHOR

Leroy Quet, Dec 04 2007

EXTENSIONS

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

STATUS

approved

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Last modified July 4 11:02 EDT 2022. Contains 355075 sequences. (Running on oeis4.)