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A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n. 28
1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 4, 2, 1, 1, 1, 1, 22, 7, 3, 2, 1, 1, 1, 1, 30, 8, 4, 2, 1, 1, 1, 1, 1, 42, 12, 5, 3, 2, 1, 1, 1, 1, 1, 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1, 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1, 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 135, 34, 13 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,g) is also the number of not necessarily connected 2-regular graphs with girth at least g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Tilman Piesk, Table for n = 1..30, table for n = 2..150 without values 1, illustrations of columns n = 2, 3, 4, 5, 6, 7, 8

FORMULA

T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.

G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic, Jun 22 2003

T(n, k) = T(n, k+1) + T(n-k, k), T(n, k) = 1 if n/2 < k <= n. - Franklin T. Adams-Watters, Jan 24 2005; Tilman Piesk, Feb 20 2016

T(n, k) = A000041(n..n-t) * transpose(A231599(k-1, 0..t)) with t = A000217(k-1). - Tilman Piesk, Feb 20 2016

Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson, Jan 31 2008

EXAMPLE

Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) = y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...

Triangle starts: - Jason Kimberley, Feb 05 2012

1;

2, 1;

3, 1, 1;

5, 2, 1, 1;

7, 2, 1, 1, 1;

11, 4, 2, 1, 1, 1;

15, 4, 2, 1, 1, 1, 1;

22, 7, 3, 2, 1, 1, 1, 1;

30, 8, 4, 2, 1, 1, 1, 1, 1;

42, 12, 5, 3, 2, 1, 1, 1, 1, 1;

56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;

77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;

101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;

From Tilman Piesk, Feb 20 2016: (Start)

n = 12, k = 4, t = A000217(k-1) = 6

vp = A000041(n..n-t) = A000041(12..6) = (77, 56, 42, 30, 22, 15, 11)

vc = A231599(k-1, 0..t) = A231599(3, 0..6) = (1,-1,-1, 0, 1, 1,-1)

T(12, 4) = vp * transpose(vc) = 77-56-42+22+15-11 = 5

(End)

MAPLE

T:= proc(n, k) option remember;

`if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))

end:

seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 28 2012

MATHEMATICA

T[n_, k_] := T[n, k] = If[ k<1 || k>n, 0, If[n == k, 1, T[n, k+1] + T[n-k, k]]]; Table [Table[ T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

PROG

(Haskell)

import Data.List (tails)

a026807 n k = a026807_tabl !! (n-1) !! (k-1)

a026807_row n = a026807_tabl !! (n-1)

a026807_tabl = map

(\row -> map (p $ last row) $ init $ tails row) a002260_tabl

where p 0 _ = 1

p _ [] = 0

p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks

-- Reinhard Zumkeller, Dec 01 2012

(Python)

from see_there import a231599_row # A231599

from sympy.ntheory import npartitions # A000041

def a026807(n, k):

if k > n:

return 0

elif k > n/2:

return 1

else:

vc = a231599_row(k-1)

t = len(vc)

vp_range = range(n-t, n+1)

vp_range = vp_range[::-1] # reverse

r = 0

for i in range(0, t):

r += vc[i] * npartitions(vp_range[i])

return r

# Tilman Piesk, Feb 21 2016

CROSSREFS

Row sums give A046746.

Cf. A026835.

Cf. A026794.

Cf. A231599.

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: this sequence (triangle); columns of this sequence: 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 simple graphs with girth exactly g [partitions with smallest part g]: A026794 (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

Cf. A002260.

Sequence in context: A210765 A160183 A168534 * A338440 A179045 A106740

Adjacent sequences: A026804 A026805 A026806 * A026808 A026809 A026810

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 5 18:47 EST 2023. Contains 360087 sequences. (Running on oeis4.)