The On-Line Encyclopedia of Integer Sequences (OEIS)

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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).
(history; published version)

#17 by Alois P. Heinz at Fri Jan 15 15:42:54 EST 2016

STATUS

proposed

approved

#16 by Jean-François Alcover at Fri Jan 15 15:15:21 EST 2016

STATUS

editing

proposed

#15 by Jean-François Alcover at Fri Jan 15 15:15:14 EST 2016

MATHEMATICA

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 *)

STATUS

approved

editing

#14 by R. J. Mathar at Mon Dec 14 03:53:48 EST 2015

STATUS

editing

approved

#13 by R. J. Mathar at Mon Dec 14 03:53:31 EST 2015

COMMENTS

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

CROSSREFS

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

STATUS

approved

editing

#12 by Alois P. Heinz at Sun Nov 29 21:11:37 EST 2015

STATUS

proposed

approved

#11 by Emeric Deutsch at Sun Nov 29 21:02:43 EST 2015

STATUS

editing

proposed

Discussion

Sun Nov 29

21:11

Alois P. Heinz: Thanks.

#10 by Emeric Deutsch at Sun Nov 29 21:01:23 EST 2015

COMMENTS

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].

STATUS

approved

editing

Discussion

Sun Nov 29

21:02

Emeric Deutsch: Added by the author on Nov 29 2015.

#9 by Alois P. Heinz at Sun Nov 29 16:27:31 EST 2015

STATUS

editing

approved

#8 by Alois P. Heinz at Sun Nov 29 16:27:18 EST 2015

LINKS

Alois P. Heinz, <a href="/A264402/b264402.txt">Rows n = 1..350, flattened</a>

MAPLE

# 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