A105552 Irregular triangle T(n,k) read down columns: the number of compositions c of n with largest_part(c)+length(c)=k+1 in row n, column k.
1, 2, 4, 1, 7, 5, 2, 11, 14, 12, 5, 1, 16, 30, 39, 32, 18, 7, 2, 22, 55, 95, 113, 101, 71, 41, 18, 6, 1, 29, 91, 195, 299, 357, 350, 292, 207, 126, 64, 27, 9, 2, 37, 140, 357, 664, 978, 1204, 1283, 1198, 992, 731, 482, 284, 148, 66, 25, 7, 1, 46, 204, 602, 1309, 2274, 3329, 4253
Offset: 1
Examples
The row n=7 starts from the partitions (weights in parentheses) 7 (1), 6+1 (2), 5+2 (2), 4+3 (2), 5+1+1 (3), 4+2+1 (6=3!/1!/1!/1!), 3+3+1 (3), 3+2+2 (3), 4+1+1+1 (4=4!/1!/3!), 3+2+1+1 (12 = 4!/1!/1!/2!), 2+2+2+1 (4), 3+1+1+1+1+1 (5), 2+2+1+1+1 (10=5!/2!/3!), 2+1+1+1+1 (6), 1+1+1+1+1+1 (1). Then T(7,7) = 1+2+3+4+5+6+1 = 22 is the sum of the weights of partitions with largest part 7 and length 1, largest part 6 and length 2,... largest part 1 and length 7. T(7,6) = 2+6+12+10 = 30 is the sum of the weights of the partitions with largest part 6 and length 1, largest part 5 and length 2, ..., largest part 1 and length 6. T(7,5) = 2+3+3+4 = 12 collects all the partitions with largest part 5 and length 1 down to largest part 1 and length 5. The array has A033638(k) nonzero entries per column, starting at n=1 as : 1 ..2 ....4 ....1..7 .......5..11 .......2..14..16 ..........12..30..22 ...........5..39..55..29 ...........1..32..95..91..37 ..............18.113.195.140 ...............7.101.299.357 ...............2
Links
- Alois P. Heinz, Columns k = 1..40, flattened
Crossrefs
Programs
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Maple
A033638 := proc(n) ( (7+(-1)^n)/2 + n^2 )/4 ; end proc: freq := proc(L,n) local a,p; a := 0 ; for p in L do if p = n then a := a+1 ; end if; end do: a ; end proc: M3 := proc(L) local a,i; a := factorial(nops(L)) ; for i in convert(L,set) do a := a/factorial(freq(L,i)) ; end do: a ; end proc: A105552 := proc(n,k) local p,a,l ; a := 0 ; for p in combinat[partition](n) do if max(op(p)) + nops(p) = k+1 then a := a+ M3(p); end if; end do ; a ; end proc: for k from 1 to 15 do for n from k to k+A033638(k)+1 do T := A105552(n,k) ; if T >0 then printf("%d,", A105552(n,k)) ; end if; end do: printf("\n") ; end do: # R. J. Mathar, Jun 26 2010 # second Maple program: b:= proc(n, k, p) option remember; `if`(n=0 and k=0, 1, `if`(k<1, 0, add(b(n-j, k-1-max(p, j)+p, max(p, j)), j=1..n))) end: T:= k-> seq(b(n, k+1, 0), n=k..k+floor((k-1)^2/4)): seq(T(k), k=1..10); # Alois P. Heinz, Jul 24 2013
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Mathematica
b[n_, k_, p_] := b[n, k, p] = If[n == 0 && k == 0, 1, If[k < 1, 0, Sum[b[n - j, k - 1 - Max[p, j] + p, Max[p, j]], {j, 1, n}]]]; T[k_] := Table[b[n, k + 1, 0], {n, k, k + Floor[(k - 1)^2/4]}]; Table[T[k], {k, 1, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
Formula
Row sums: Sum_{k=0..n} T(n,k) = 2^(n-1).
Column sums: Sum_{n>=k} T(n,k) = A047970(n).
Extensions
Definition clarified by R. J. Mathar, Jun 26 2010
Comments