A004250 -id:A004250 - cp's OEIS frontend

A347543 Number of partitions of n into 7 or more parts.

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 134, 186, 255, 345, 461, 611, 801, 1043, 1346, 1727, 2199, 2787, 3508, 4398, 5482, 6809, 8414, 10365, 12711, 15545, 18935, 23006, 27854, 33646, 40513, 48680, 58326, 69748, 83192, 99048, 117650, 139513, 165083, 195034, 229968, 270760

Offset: 7

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008636, A035300, A035301, A347542, A347544, A347545, A347547.

nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 7, nmax}], {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: Sum_{k>=7} x^k / Product_{j=1..k} (1 - x^j).

A347544 Number of partitions of n into 8 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 96, 137, 190, 263, 356, 480, 637, 842, 1098, 1427, 1835, 2351, 2986, 3780, 4749, 5949, 7405, 9190, 11344, 13966, 17111, 20913, 25454, 30908, 37393, 45141, 54315, 65222, 78090, 93317, 111220, 132323, 157050, 186088, 220015, 259716

Offset: 8

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008637, A035300, A035301, A347542, A347543, A347545, A347547.

nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 8, nmax}], {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: Sum_{k>=8} x^k / Product_{j=1..k} (1 - x^j).

A347545 Number of partitions of n into 9 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 138, 193, 267, 364, 491, 656, 868, 1139, 1483, 1917, 2461, 3142, 3985, 5030, 6315, 7893, 9817, 12165, 15007, 18451, 22597, 27589, 33565, 40724, 49249, 59410, 71460, 85753, 102632, 122574, 146032, 173638, 206003, 243951, 288296, 340124

Offset: 9

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008638, A035300, A035301, A347542, A347543, A347544, A347547.

nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 9, nmax}], {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: Sum_{k>=9} x^k / Product_{j=1..k} (1 - x^j).

A347547 Number of partitions of n into 10 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 270, 368, 499, 667, 887, 1165, 1524, 1973, 2544, 3253, 4143, 5239, 6602, 8268, 10320, 12813, 15859, 19537, 24000, 29359, 35820, 43541, 52795, 63803, 76929, 92476, 110926, 132694, 158414, 188649, 224231, 265916, 314793

Offset: 10

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008639, A035300, A035301, A347542, A347543, A347544, A347545.

nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 10, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: Sum_{k>=10} x^k / Product_{j=1..k} (1 - x^j).

A029894 Number of directed (or Gale-Ryser) graphical partitions: degree-vector pairs (in-degree, out-degree) for directed graphs (loops allowed) with n vertices; or possible ordered pair (row-sum, column-sum) vectors for a 0-1 matrix.

Original entry on oeis.org

1, 2, 7, 34, 221, 1736, 15584, 153228, 1611189, 17826202, 205282376, 2441437708, 29816628471, 372314544202, 4737438631001, 61264426341926, 803488037899349, 10668478221202710, 143203795004873285, 1940953294927992976, 26536578116407809962, 365653739580163294032

Offset: 0

torsten.sillke(AT)lhsystems.com

nonn

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Andrew Howroyd, Table of n, a(n) for n = 0..30
Peter L. Erdos, I Miklós, Z Toroczkai, New classes of degree sequences with fast mixing swap Markov chain sampling, arXiv preprint arXiv:1601.08224 [math.CO], 2016.
Index entries for sequences related to graphical partitions

Main diagonal of A327913.

Cf. A000569, A004250, A004251, A029889, A318396.

\\ see A327913 for T(n,m)
for(n=0, 15, print1(T(n,n), ", ")) \\ Andrew Howroyd, Nov 01 2019

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

a(n) = F(n, n, 0, n) where F(b, c, t, w) = Sum_{i=0..b} Sum_{j=ceiling((t+i)/w)..min(t+i, c)} F(i, j, t+i-j, w-1) for w > 0, F(b, c, 0, 0) = 1 and F(b, c, t, 0) = 0 for t > 0. - Andrew Howroyd, Nov 01 2019

"Loops allowed" added to the definition by Brendan McKay, Oct 20 2015

a(0)=1 prepended and terms a(12) and beyond from Andrew Howroyd, Oct 31 2019

A283826 Irregular triangle read by rows: T(n,k) = number of trees on n nodes with radius k, n>=1, 1 <= k <= floor(n/2).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 3, 1, 11, 10, 1, 1, 17, 25, 4, 1, 25, 61, 18, 1, 1, 36, 132, 61, 5, 1, 50, 277, 194, 28, 1, 1, 70, 554, 553, 117, 6, 1, 94, 1077, 1495, 451, 40, 1, 1, 127, 2034, 3823, 1552, 197, 7, 1, 168, 3770, 9427, 5020, 879, 54, 1, 1, 222, 6853, 22466, 15289, 3485, 305, 8, 1

Offset: 1

N. J. A. Sloane, Mar 19 2017

The radius of a tree is the maximal distance of a node from the center.

			Triangle begins:
  0,
  1,
  1,
  1,  1,
  1,  2,
  1,  4,   1,
  1,  7,   3,
  1, 11,  10,   1,
  1, 17,  25,   4,
  1, 25,  61,  18,  1,
  1, 36, 132,  61,  5,
  1, 50, 277, 194, 28, 1,
  ...

Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Cf. A283827.

See also A000676, A000677, A027416, A102911, A004250 (column 2?), A000055 (row sums).

T(n,k) = A034853(n,2k-1) + A034853(n,2k). - R. J. Mathar, Apr 03 2017

A007722 Number of graphical partitions of biconnected graphs with n nodes.

Original entry on oeis.org

1, 3, 9, 34, 125, 473, 1779, 6732, 25492, 96927, 369463, 1412700, 5415117, 20807502, 80120350, 309106496, 1194609429, 4624160156, 17925278497, 69578272204, 270401326899, 1052036082719, 4097343156323, 15973179953261, 62325892264031, 243392644741599

Offset: 3

Frank Ruskey

nonn

F. Ruskey, Alley CATs in search of good homes, Congress. Numerant., 102 (1994) 97-110.

Wang Kai, Table of n, a(n) for n = 3..101
Kai Wang, Efficient Counting of Degree Sequences, arXiv preprint arXiv:1604.04148 [math.CO], 2016-2017.
Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A007721, A029889, A095268.

a(15)-a(28) added by Kai Wang, Feb 15 2017

A029890 Number of odd graphical partitions.

Original entry on oeis.org

1, 2, 7, 20, 70, 234, 832, 2956, 10759, 39394, 145892, 543564, 2038831, 7684116, 29092055, 110550260, 421495147, 1611662256

Offset: 1

TORSTEN.SILLKE(AT)LHSYSTEMS.COM

nonn

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

A029891 Number of even graphical partitions.

Original entry on oeis.org

1, 3, 7, 23, 70, 242, 832, 2983, 10759, 39482, 145892, 543877, 2038831, 7685211, 29092055, 110554267, 421495147, 1611676767

Offset: 1

TORSTEN.SILLKE(AT)LHSYSTEMS.COM

nonn

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

A259873 Triangle read by rows: T(n,k) (n >= 3, 3 <= k <= n) = number of possible graphical partitions for simple graphs with n non-isolated nodes and k edges.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 7, 0, 0, 4, 9, 11, 0, 0, 2, 11, 15, 17, 0, 0, 1, 11, 22, 25, 25, 0, 0, 1, 9, 26, 38, 37, 36, 0, 0, 0, 7, 29, 49, 58, 55, 50, 0, 0, 0, 5, 29, 63, 81, 87, 77, 70

Offset: 3

N. J. A. Sloane, Jul 09 2015

			Triangle begins:
1,
0,2,
0,1,4,
0,1,4,7,
0,0,4,9,11,
0,0,2,11,15,17,
0,0,1,11,22,25,25,
0,0,1,9,26,38,37,36,
0,0,0,7,29,49,58,55,50,
0,0,0,5,29,63,81,87,77,70,
...

P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). Contains table for n <= 27.

P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]

A004250 is a diagonal. Cf. A000088, A004251.

cp's OEIS Frontend

A347543 Number of partitions of n into 7 or more parts.

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A347544 Number of partitions of n into 8 or more parts.

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A347545 Number of partitions of n into 9 or more parts.

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A347547 Number of partitions of n into 10 or more parts.

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A029894 Number of directed (or Gale-Ryser) graphical partitions: degree-vector pairs (in-degree, out-degree) for directed graphs (loops allowed) with n vertices; or possible ordered pair (row-sum, column-sum) vectors for a 0-1 matrix.

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A283826 Irregular triangle read by rows: T(n,k) = number of trees on n nodes with radius k, n>=1, 1 <= k <= floor(n/2).

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A007722 Number of graphical partitions of biconnected graphs with n nodes.

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A029890 Number of odd graphical partitions.

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A029891 Number of even graphical partitions.

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A259873 Triangle read by rows: T(n,k) (n >= 3, 3 <= k <= n) = number of possible graphical partitions for simple graphs with n non-isolated nodes and k edges.

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