A026794 - cp's OEIS frontend

A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1, 22, 4, 2, 1, 0, 0, 0, 0, 1, 30, 7, 2, 1, 1, 0, 0, 0, 0, 1, 42, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Clark Kimberling

At least one part is k and each part is at least k.
From Emeric Deutsch, Feb 19 2006: (Start)
Also number of partitions of n in which the largest part occurs exactly k times. Example: T(6,2)=2 because we have [3,3] and [2,2,1,1].
G.f. of column k is x^k/prod(j>=k, 1-x^j ) (k>=1).
Row sums yield the partition numbers (A000041).
T(n,1) = A000041(n-1) (the partition numbers).
T(n,2) = A002865(n-2) (n>=2).
T(n,3)=A026796(n). T(n,4) = A026797(n). T(n,5) = A026798(n). T(n,6) = A026799(n). T(n,7) = A026800(n). T(n,8) = A026801(n). T(n,9) = A026802(n). T(n,10) = A026803(n).
Sum(k*T(n,k),k=1..n) = A046746(n). (End)
Triangle inverse = A161363. - Gary W. Adamson, Jun 07 2009
T(n,g) is also the number of not necessarily connected 2-regular graphs with girth exactly g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012
From Bob Selcoe, Jul 24 2014 (Start):
Below is a process to generate equations for column k.
Let P be the partition numbers A000041(n-j) and let f(k) denote equations which generate column k.
To find f(k), start with f(1) = P(n-j), j=1. Thus T(n,1) = f(1) = P(n-1). This is the equation for column 1.
To find f(k) k>1, first sum the terms of f(k-1) replacing the value j with j+1, and then subtract the terms of f(k-1) replacing the value j with j+k. So to find f(2) (i.e., the equation for column 2, where k=2), start with f(1) = P(n-1); first replace j with j+1 (yielding P(n-2)), and then replace j with j+2 (yielding P(n-3)).  Subtracting the second term from the first, we get: f(2) = P(n-2) - P(n-3).
To find f(3), start with f(2), replace j with j+1 (yielding P(n-3) - P(n-4)) and then replace j with j+3 (yielding P(n-5) - P(n-6)). Subtracting the second group of terms from the first, we get: f(3) = P(n-3) - P(n-4) - P(n-5) + P(n-6). This is the equation for column 3; also the equation for T(n,3) = A026796(n). So for example, T(13,3) = 5 because P(13-3) - P(13-4) - P(13-5) + P(13-6) = 42 - 30 - 22 + 15 = 5.
Continue as above to find f(k) k={4..inf.}.  This will generate equations for T(n,4) = A026797(n), T(n,5) = A026798(n), T(n,6) = A026799(n), ad inf.
(End)

			T(12,3) = 4 because we have [9,3], [6,3,3], [5,4,3] and [3,3,3,3]. - Edited by _Bob Selcoe_, Sep 03 2016
Triangle starts:
    1;
    1,  1;
    2,  0, 1;
    3,  1, 0, 1;
    5,  1, 0, 0, 1;
    7,  2, 1, 0, 0, 1;
   11,  2, 1, 0, 0, 0, 1;
   15,  4, 1, 1, 0, 0, 0, 1;
   22,  4, 2, 1, 0, 0, 0, 0, 1;
   30,  7, 2, 1, 1, 0, 0, 0, 0, 1;
   42,  8, 3, 1, 1, 0, 0, 0, 0, 0, 1;
   56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1;
   77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  135, 24, 9, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Kevin Brown, On Euler's Pentagonal Theorem, 1994-2008.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g.
Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No. 1, December 2008. pp. 176-187.
Tilman Piesk, First 50 rows and illustrations of columns n = 2, 3, 4, 5, 6, 7, 8. a(n, k) is the number of gray fields in row n of table k.

Row sums give A000041.
Cf. A000041, A046746, A008284, A161363, A161364, A238341.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: this sequence (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012

g:=sum(t^i*x^i/product(1-x^j,j=i..30),i=1..30): gser:=simplify(series(g,x=0,19)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # Emeric Deutsch, Feb 19 2006
nmax:=13; for n from 1 to nmax do T(n,n):=1 od: for n from 1 to nmax do for k from floor(n/2)+1 to n-1 do T(n,k):=0 od: od: for n from 2 to nmax do for k from 1 to floor(n/2) do T(n,k):=sum(T(n-k,i),i=k..n-k) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
nmax:=13; with(combinat): for n from 1 to nmax do for k from n+1 to nmax do T(n,k):=0 od: od: for n from 1 to nmax do T(n,1):=numbpart(n-1) od: for n from 1 to nmax do T(n,n):=1 od: for n from 2 to nmax do for k from 2 to n-1 do T(n,k) := T(n-1,k-1) - T(n-k,k-1) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
#
p:= (f, g)-> zip((x,y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local h;
      h:= `if`(n=i and i>0, [0$(i-1), 1], []);
      `if`(i<1, h, p(p(h, b(n, i-1)), `if`(n b(n, n)[]:
seq(T(n), n=1..14); # Alois P. Heinz, Mar 28 2012

t[n_, k_] /; k<1 || k>n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[t[n-k, i], {i, k, n-k}]; Flatten[ Table[t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, May 11 2012, after PARI *)

{T(n, k) = if( k<1 || k>n, 0, if( n==k, 1, sum(i=k, n-k, T(n-k, i))))} \\ Michael Somos, Feb 06 2003

A026794(n,k)=#select(p->p[1]==k,partitions(n,[k,n])) \\ For illustration: Creates the list of all partitions of n with smallest part equal to k. - M. F. Hasler, Jun 14 2018

T(n, k) = sum{T(n-k, i), k<=i<=n-k} for k=1, 2, ..., m, T(n, k)=0 for k=m+1, ..., n-1, where m=floor(n/2); T(n, n)=1 for n >= 1.
G.f.: G(t,x)=sum(t^i*x^i/product(1-x^j, j=i..infinity), i=1..infinity). - Emeric Deutsch, Feb 19 2006
G.f.: Sum_{k>=1} tx^k/(1-tx^k)/product(1-x^j,j=1..k-1). - Emeric Deutsch, Mar 13 2006
T(n,k) = T(n-1,k-1) - T(n-k,k-1) for n>=2 and 2<=k<=(n-1) with T(n,1) = A000041(n-1), T(n,n) = 1 for n>=1 and T(n,k) = 0 for k>n. - Johannes W. Meijer, Jun 21 2010
T(k,k) = 1 and T(n,1) = row sum (n-1); thus Meijer's 2010 formula generates the triangle without a priori reference to A000041 (the partition sequence). - Bob Selcoe, Sep 03 2016

More terms from Emeric Deutsch, Feb 19 2006

cp's OEIS Frontend

A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

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