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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.

%I A026807 #61 Oct 18 2019 07:41:23
%S A026807 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,
%T A026807 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,
%U A026807 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
%N 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.
%C A026807 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
%H A026807 Alois P. Heinz, <a href="/A026807/b026807.txt">Rows n = 1..141, flattened</a>
%H A026807 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_ge_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g</a>
%H A026807 Tilman Piesk, <a href="http://paste.watchduck.net/1602/intpart/A026807_full.html">Table for n = 1..30</a>, <a href="http://paste.watchduck.net/1602/intpart/A026807_half.html">table for n = 2..150 without values 1</a>, illustrations of columns n = <a href="http://paste.watchduck.net/1602/intpart/addends_ge_2.html">2</a>, <a href="http://paste.watchduck.net/1602/intpart/addends_ge_3.html">3</a>, <a href="http://paste.watchduck.net/1602/intpart/addends_ge_4.html">4</a>, <a href="http://paste.watchduck.net/1602/intpart/addends_ge_5.html">5</a>, <a href="http://paste.watchduck.net/1602/intpart/addends_ge_6.html">6</a>, <a href="http://paste.watchduck.net/1602/intpart/addends_ge_7.html">7</a>, <a href="http://paste.watchduck.net/1602/intpart/addends_ge_8.html">8</a>
%F A026807 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.
%F A026807 G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - _Vladeta Jovovic_, Jun 22 2003
%F A026807 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
%F A026807 T(n, k) = A000041(n..n-t) * transpose(A231599(k-1, 0..t)) with t = A000217(k-1). - _Tilman Piesk_, Feb 20 2016
%F A026807 Equals A026794 * A000012 as infinite lower triangular matrices. - _Gary W. Adamson_, Jan 31 2008
%e A026807 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+...
%e A026807 Triangle starts:  - _Jason Kimberley_, Feb 05 2012
%e A026807 1;
%e A026807 2, 1;
%e A026807 3, 1, 1;
%e A026807 5, 2, 1, 1;
%e A026807 7, 2, 1, 1, 1;
%e A026807 11, 4, 2, 1, 1, 1;
%e A026807 15, 4, 2, 1, 1, 1, 1;
%e A026807 22, 7, 3, 2, 1, 1, 1, 1;
%e A026807 30, 8, 4, 2, 1, 1, 1, 1, 1;
%e A026807 42, 12, 5, 3, 2, 1, 1, 1, 1, 1;
%e A026807 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;
%e A026807 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;
%e A026807 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
%e A026807 From _Tilman Piesk_, Feb 20 2016: (Start)
%e A026807 n = 12, k = 4, t = A000217(k-1) = 6
%e A026807 vp = A000041(n..n-t) = A000041(12..6) = (77, 56, 42, 30, 22, 15, 11)
%e A026807 vc = A231599(k-1, 0..t) = A231599(3, 0..6) = (1,-1,-1, 0, 1, 1,-1)
%e A026807 T(12, 4) = vp * transpose(vc) = 77-56-42+22+15-11 = 5
%e A026807 (End)
%p A026807 T:= proc(n, k) option remember;
%p A026807       `if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))
%p A026807     end:
%p A026807 seq(seq(T(n, k), k=1..n), n=1..14); # _Alois P. Heinz_, Mar 28 2012
%t A026807 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_ *)
%o A026807 (Haskell)
%o A026807 import Data.List (tails)
%o A026807 a026807 n k = a026807_tabl !! (n-1) !! (k-1)
%o A026807 a026807_row n = a026807_tabl !! (n-1)
%o A026807 a026807_tabl = map
%o A026807    (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
%o A026807    where p 0  _ = 1
%o A026807          p _ [] = 0
%o A026807          p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
%o A026807 -- _Reinhard Zumkeller_, Dec 01 2012
%o A026807 (Python)
%o A026807 from see_there import a231599_row  # A231599
%o A026807 from sympy.ntheory import npartitions  # A000041
%o A026807 def a026807(n, k):
%o A026807     if k > n:
%o A026807         return 0
%o A026807     elif k > n/2:
%o A026807         return 1
%o A026807     else:
%o A026807         vc = a231599_row(k-1)
%o A026807         t = len(vc)
%o A026807         vp_range = range(n-t, n+1)
%o A026807         vp_range = vp_range[::-1]  # reverse
%o A026807         r = 0
%o A026807         for i in range(0, t):
%o A026807             r += vc[i] * npartitions(vp_range[i])
%o A026807         return r
%o A026807 # _Tilman Piesk_, Feb 21 2016
%Y A026807 Row sums give A046746.
%Y A026807 Cf. A026835.
%Y A026807 Cf. A026794.
%Y A026807 Cf. A231599.
%Y A026807 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
%Y A026807 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
%Y A026807 Cf. A002260.
%K A026807 nonn,tabl
%O A026807 1,2
%A A026807 _Clark Kimberling_