Displaying 1-10 of 39 results found.
Prune the tree structure defined in sequence A129129 yielding a new irregular table with shape sequence A027336.
+20
0
1, 2, 3, 5, 6, 7, 10, 9, 11, 14, 15, 18, 13, 22, 21, 25, 30, 27, 17, 26, 33, 35, 42, 50, 45, 54, 19, 34, 39, 55, 66, 49, 70, 63, 75, 90, 81
COMMENTS
A000041 begins 1 1 2 3 5 7 11 15 22 30 42 ... and when shifted 0 0 1 1 2 3 5 7 11 15 22 ... by subtraction yielding 1 1 1 2 3 4 6 8 11 15 20 ... shape seq A027336.
EXAMPLE
The original tree begins
1;
2;
3, 4;
5, 6, 8;
7, 10, 9, 12, 16;
11, 14, 15, 20, 18, 24, 32;
...
Multiplying by four yields
4;
8;
12, 16;
20, 24, 32;
...
These are the values to be omitted, leaving
1;
2;
3;
5, 6;
7, 10, 9;
11, 14, 15, 18;
...
Irregular triangle read by rows: T(n,k), n >= 0, k >= 1, in which if n is even then row n lists the first A008619(n) even indexed terms of A027336 otherwise if n is odd then row n lists the first A008619(n) odd indexed terms of A027336.
+20
0
1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 8, 1, 1, 3, 6, 11, 1, 2, 4, 8, 15, 1, 1, 3, 6, 11, 20, 1, 2, 4, 8, 15, 26, 1, 1, 3, 6, 11, 20, 35, 1, 2, 4, 8, 15, 26, 45, 1, 1, 3, 6, 11, 20, 35, 58, 1, 2, 4, 8, 15, 26, 45, 75, 1, 1, 3, 6, 11, 20, 35, 58, 96, 1, 2, 4, 8, 15, 26, 45, 75, 121
COMMENTS
The sum of row n equals the number of partitions of n.
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 2;
1, 1, 3;
1, 2, 4;
1, 1, 3, 6;
1, 2, 4, 8;
1, 1, 3, 6, 11;
1, 2, 4, 8, 15;
1, 1, 3, 6, 11, 20;
1, 2, 4, 8, 15, 26;
1, 1, 3, 6, 11, 20, 35;
1, 2, 4, 8, 15, 26, 45;
1, 1, 3, 6, 11, 20, 35, 58;
1, 2, 4, 8, 15, 26, 45, 75;
1, 1, 3, 6, 11, 20, 35, 58, 96;
1, 2, 4, 8, 15, 26, 45, 75, 121;
...
For n = 10 the sum of the 10th row is 1 + 1 + 3 + 6 + 11 + 20 = 42, the same as the number of partitions of 10.
Number of partitions of n that do not contain 1 as a part.
(Formerly M0309 N0113)
+10
375
1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701
COMMENTS
Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - Franklin T. Adams-Watters, May 01 2006 Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011 From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)
a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing ( A053871), that are unique under permutations preserving the given pairing. Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)
Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = A058682 starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle A171239 = A058682. - Gary W. Adamson, Dec 05 2009 Also the number of 2-regular multigraphs with loops forbidden. - Jason Kimberley, Jan 05 2011 Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011
Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). - N. J. A. Sloane, Sep 16 2013 The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]_q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - William J. Keith, Nov 14 2013 Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - Lyndsey Wong, Jul 09 2016 The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016 Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - R. J. Mathar, Mar 25 2019 For n > 1, also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - Spencer Miller, Jan 30 2023] Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:
1: (1)
2: (2)
3: (3) or (21)
4: (22) or (211)
5: (32) or (221)
6: (2211)
7: (322)
8: (3221)
9: (32211)
(End)
There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - Andrew Yezhou Wang, Dec 14 2019 Let P(2, n) denote the set of partitions of n into parts k > 1. Then A000041(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - A000041(5). (End)
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
G.f.: Product_{m>1} 1/(1-x^m).
a(0)=1, a(n) = p(n) - p(n-1), n >= 1, with the partition numbers p(n) := A000041(n). G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - Joerg Arndt, Apr 13 2011 G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - Joerg Arndt, Apr 17 2011 a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Feb 26 2015, extended Nov 04 2016 G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018 G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).
More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)
G.f.: 1 + Sum_{n >= 1} x^(n+1)/Product_{k = 1..n-1} 1 - x^(k+2). - Peter Bala, Dec 01 2024
EXAMPLE
a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...
The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408. (2) (3) (4) (5) (6) (7) (8) (9)
(22) (32) (33) (43) (44) (54)
(42) (52) (53) (63)
(222) (322) (62) (72)
(332) (333)
(422) (432)
(2222) (522)
(3222)
The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by A247180. (1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(41) (51) (52) (62)
(221) (321) (61) (71)
(331) (332)
(421) (431)
(2221) (521)
(3221)
The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by A325761. (21) (22) (32) (42) (52) (62) (72) (82)
(311) (321) (322) (332) (333) (433)
(331) (431) (432) (532)
(4111) (4211) (531) (631)
(4221) (4222)
(4311) (4321)
(51111) (4411)
(52111)
The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by A070003. (11) (111) (22) (221) (33) (331) (44)
(1111) (11111) (222) (2221) (332)
(2211) (22111) (2222)
(111111) (1111111) (3311)
(22211)
(221111)
(11111111)
Nonisomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.
{12,12} {12,13,23} {12,12,34,34} {12,12,34,35,45} {12,12,34,34,56,56}
{12,13,24,34} {12,13,24,35,45} {12,12,34,35,46,56}
{12,13,23,45,46,56}
{12,13,24,35,46,56}
The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by A325762. (11) (111) (211) (2111) (2211) (22111) (22211) (33111)
(1111) (11111) (3111) (31111) (32111) (222111)
(21111) (211111) (41111) (321111)
(111111) (1111111) (221111) (411111)
(311111) (2211111)
(2111111) (3111111)
(11111111) (21111111)
(111111111)
(End)
MAPLE
with(combstruct): ZL1:=[S, {S=Set(Cycle(Z, card>1))}, unlabeled]: seq(count(ZL1, size=n), n=0..50); # Zerinvary Lajos, Sep 24 2007 G:= {P=Set (Set (Atom, card>1))}: combstruct[gfsolve](G, unlabeled, x): seq (combstruct[count] ([P, G, unlabeled], size=i), i=0..50); # Zerinvary Lajos, Dec 16 2007 with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50); # Zerinvary Lajos, Jun 11 2008 # alternative Maple program:
A002865:= proc(n) option remember; `if`(n=0, 1, add(
(numtheory[sigma](j)-1)* A002865(n-j), j=1..n)/n) end:
MATHEMATICA
Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* Robert G. Wilson v, Jul 24 2004 *) f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* Robert G. Wilson v *) Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Aug 18 2018 *) CoefficientList[Series[1/QPochhammer[x^2, x], {x, 0, 50}], x] (* G. C. Greubel, Nov 03 2019 *) Table[Count[IntegerPartitions[n], _?(FreeQ[#, 1]&)], {n, 0, 50}] (* Harvey P. Dale, Feb 12 2023 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))};
(Magma) A41 := func<n|n ge 0 select NumberOfPartitions(n) else 0>; [A41(n)-A41(n-1):n in [0..50]]; // Jason Kimberley, Jan 05 2011 (GAP) Concatenation([1], List([1..41], n->NrPartitions(n)-NrPartitions(n-1))); # Muniru A Asiru, Aug 20 2018 (SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()
(Python)
from sympy import npartitions
def A002865(n): return npartitions(n)-npartitions(n-1) if n else 1 # Chai Wah Wu, Mar 30 2023
CROSSREFS
Pairwise sums seem to be in A027336. 2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - Jason Kimberley, Jan 05 2011 See also A098743 (parts that do not divide n). Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: A274161. - Lyndsey Wong, Jul 09 2016 Row sums of characteristic array A145573.
Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.
+10
132
1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 3, 0, 1, 2, 0, 0, 0, 0, 1, 4, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 6, 1, 3, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 8, 1, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 11, 2, 7, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
Triangle begins:
1
1 0 1
1 1 0 0 1
2 0 2 0 0 0 1
3 0 1 2 0 0 0 0 1
4 1 2 0 3 0 0 0 0 0 1
6 1 3 0 1 3 0 0 0 0 0 0 1
8 1 6 0 2 0 4 0 0 0 0 0 0 0 1
11 2 7 1 3 0 1 4 0 0 0 0 0 0 0 0 1
15 2 10 3 4 0 2 0 5 0 0 0 0 0 0 0 0 0 1
20 3 13 3 7 0 3 0 1 5 0 0 0 0 0 0 0 0 0 0 1
26 4 19 3 11 1 4 0 2 0 6 0 0 0 0 0 0 0 0 0 0 0 1
For example, row n = 8 counts the following partitions:
611 4211 422 . 332 . 44 . . . . . . . 8
5111 521 431 53
32111 2222 62
41111 3221 71
221111 3311
311111 22211
2111111
11111111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[#]==k&]], {n, 1, 10}, {k, 1, n, 1/2}]
CROSSREFS
Row lengths are 2n-1 = A005408(n-1). The median statistic is ranked by A360005(n)/2. A240219 counts partitions w/ the same mean as median, complement A359894.
Triangle read by rows where T(n,k) is the number of integer partitions of n with median k = 1..n.
+10
97
1, 1, 1, 1, 0, 1, 2, 2, 0, 1, 3, 1, 0, 0, 1, 4, 2, 3, 0, 0, 1, 6, 3, 1, 0, 0, 0, 1, 8, 6, 2, 4, 0, 0, 0, 1, 11, 7, 3, 1, 0, 0, 0, 0, 1, 15, 10, 4, 2, 5, 0, 0, 0, 0, 1, 20, 13, 7, 3, 1, 0, 0, 0, 0, 0, 1, 26, 19, 11, 4, 2, 6, 0, 0, 0, 0, 0, 1
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
Triangle begins:
1
1 1
1 0 1
2 2 0 1
3 1 0 0 1
4 2 3 0 0 1
6 3 1 0 0 0 1
8 6 2 4 0 0 0 1
11 7 3 1 0 0 0 0 1
15 10 4 2 5 0 0 0 0 1
20 13 7 3 1 0 0 0 0 0 1
26 19 11 4 2 6 0 0 0 0 0 1
35 24 14 5 3 1 0 0 0 0 0 0 1
45 34 17 8 4 2 7 0 0 0 0 0 0 1
58 42 23 12 5 3 1 0 0 0 0 0 0 0 1
For example, row n = 9 counts the following partitions:
(7,1,1) (5,2,2) (3,3,3) (4,4,1) . . . . (9)
(6,1,1,1) (6,2,1) (4,3,2)
(3,3,1,1,1) (3,2,2,2) (5,3,1)
(4,2,1,1,1) (4,2,2,1)
(5,1,1,1,1) (4,3,1,1)
(3,2,1,1,1,1) (2,2,2,2,1)
(4,1,1,1,1,1) (3,2,2,1,1)
(2,2,1,1,1,1,1)
(3,1,1,1,1,1,1)
(2,1,1,1,1,1,1,1)
(1,1,1,1,1,1,1,1,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[#]==k&]], {n, 15}, {k, n}]
CROSSREFS
Including half-steps gives A359893. The median statistic is ranked by A360005(n)/2. A240219 counts partitions w/ the same mean as median, complement A359894.
Number of partitions of n into parts >= 3.
+10
67
1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510
COMMENTS
a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011 Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011 By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3. For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message. - E. Keith Lloyd (ekl(AT)soton.ac.uk).
For n >= 1, also the number of regular graphs of degree 2. - Mitch Harris, Jun 22 2005 (1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2*x^6 + ...) = (1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ...) * 1 / (1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + ...). - Gary W. Adamson, Jun 30 2009 Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. - Jason Kimberley, Oct 05 2009 Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014 For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482. For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], [4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014 Let P(3, n) denote the set of partitions of n into parts k >= 3. Then A000041(n) = (1/2) * Sum_{parts k in all partitions in P(3, n+3)} phi(k), where phi(k) is the Euler totient function (see A000010). For example, with n = 5, there are 3 partitions of n + 3 = 8 into parts greater then 3, namely, 8, 5 + 3 and 4 + 4, and (1/2)*(phi(8) + phi(5) + phi(3) + 2*phi(4)) = 7 = A000041(5). (End)
FORMULA
a(n) = p(n) - p(n - 1) - p(n - 2) + p(n - 3) where p(n) is the number of unrestricted partitions of n into positive parts ( A000041). G.f.: Product_{m>=3} 1/(1-x^m).
G.f.: (Sum_{n>=0} x^(3*n)) / (Product_{k=1..n} (1 - x^k)). - Joerg Arndt, Apr 17 2011 a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Feb 26 2015 G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018 G.f.: 1 + Sum_{n >= 1} x^(n+2)/Product_{k = 0..n-1} (1 - x^(k+3)). - Peter Bala, Dec 01 2024
MAPLE
series(1/product((1-x^i), i=3..50), x, 51);
ZL := [ B, {B=Set(Set(Z, card>=3))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..46); # Zerinvary Lajos, Mar 13 2007 with(combstruct):ZL2:=[S, {S=Set(Cycle(Z, card>2))}, unlabeled]:seq(count(ZL2, size=n), n=0..46); # Zerinvary Lajos, Sep 24 2007 with(combstruct):a:=proc(m) [A, {A=Set(Cycle(Z, card>m))}, unlabeled]; end: A008483:=a(2):seq(count( A008483, size=n), n=0..46); # Zerinvary Lajos, Oct 02 2007
MATHEMATICA
f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 3], {n, 49}] (* Robert G. Wilson v, Jan 31 2011 *) Rest[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Length[p]]], {n, 50}]] (* Clark Kimberling, Feb 27 2014 *)
PROG
(Magma) p := NumberOfPartitions; A008483 := func< n | n eq 0 select 1 else n le 2 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3)>; // Jason Kimberley, Jan 11 2011
CROSSREFS
2-regular simple graphs: A179184 (connected), A165652 (disconnected), this sequence (not necessarily connected). 2-regular not necessarily connected graphs without multiple edges [partitions without 2 as a part]: this sequence (no loops allowed [without 1 as a part]), A027336 (loops allowed [parts may be 1]). 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), this sequence (g=3), A008484 (g=4), A185325 (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
AUTHOR
T. Forbes (anthony.d.forbes(AT)googlemail.com)
Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.
+10
33
1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 2, 1, 4, 6, 4, 1, 7, 8, 5, 2, 8, 11, 8, 3, 12, 15, 10, 4, 1, 14, 20, 15, 6, 1, 21, 26, 19, 9, 2, 24, 35, 27, 12, 3, 34, 45, 34, 17, 5, 41, 58, 47, 23, 6, 1, 55, 75, 59, 31, 10, 1, 66, 96, 79, 41, 13, 2
COMMENTS
The "least gap" or "mex" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
Sum of entries in row n is A000041(n).
FORMULA
G.f.: G(t,x) = Sum_{j>=1} (t^j*x^{j(j-1)/2}*(1-x^j))/Product_{i>=1}(1-x^i).
EXAMPLE
Row n=5 is 2,3,2; indeed, the least gaps of [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] are 1, 2, 1, 2, 3, 3, and 2, respectively (i.e., two 1s, three 2s, and two 3s).
Triangle begins:
1
0 1
1 1
1 1 1
2 2 1
2 3 2
4 4 2 1
4 6 4 1
7 8 5 2
8 11 8 3
12 15 10 4 1
14 20 15 6 1
21 26 19 9 2
MAPLE
g := (sum(t^j*x^((1/2)*j*(j-1))*(1-x^j), j = 1 .. 80))/(product(1-x^i, i = 1 .. 80)): gser := simplify(series(g, x = 0, 23)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, [1, 0],
[0, x]), `if`(i<1, 0, (p-> [0, p[2] +p[1]*x^i])(
b(n, i-1)) +add(b(n-i*j, i-1), j=1..n/i)))
end:
T:= n->(p->seq(coeff(p, x, i), i=1..degree(p)))(b(n, n+1)[2]):
MATHEMATICA
Needs["Combinatorica`"]; {1, 0}~Join~Flatten[Table[Count[Map[If[# == {}, 0, First@ #] &@ Complement[Range@ n, #] &, Combinatorica`Partitions@ n], n_ /; n == k], {n, 17}, {k, n}] /. 0 -> Nothing] (* Michael De Vlieger, Nov 21 2015 *) mingap[q_]:=Min@@Complement[Range[If[q=={}, 0, Max[q]]+1], q]; Table[Length[Select[IntegerPartitions[n], mingap[#]==k&]], {n, 0, 15}, {k, Round[Sqrt[2*(n+1)]]}] (* Gus Wiseman, Apr 19 2021 *) b[n_, i_] := b[n, i] = If[n == 0, If[i == 0, {1, 0}, {0, x}], If[i<1, {0, 0}, {0, #[[2]] + #[[1]]*x^i}&[b[n, i-1]] + Sum[b[n-i*j, i - 1], {j, 1, n/i}]]];
T[n_] := CoefficientList[b[n, n + 1], x][[2]] // Rest;
CROSSREFS
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A257993 gives the least gap of the partition with Heinz number n.
A339564 counts factorizations with a selected factor.
A342050 ranks partitions with even least gap.
A342051 ranks partitions with odd least gap.
Table read by rows: number of partitions of n with k as low median.
+10
26
1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1, 6, 3, 1, 0, 0, 1, 8, 4, 2, 0, 0, 0, 1, 11, 6, 3, 1, 0, 0, 0, 1, 15, 8, 4, 2, 0, 0, 0, 0, 1, 20, 12, 5, 3, 1, 0, 0, 0, 0, 1, 26, 16, 7, 4, 2, 0, 0, 0, 0, 0, 1, 35, 22, 10, 5, 3, 1, 0, 0, 0, 0, 0, 1, 45, 29, 14, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 58, 40, 19, 8, 5, 3, 1
COMMENTS
For a multiset with an odd number of elements, the low median is the same as the median. For a multiset with an even number of elements, the low median is the smaller of the two central elements.
Arrange the parts of a partition nonincreasing order. Remove the first part, then the last, then the first remaining part, then the last remaining part, and continue until only a single number, the low median, remains. - Clark Kimberling, May 16 2019
EXAMPLE
For the partition [2,1^2], the sole middle element is 1, so that is the low median. For [3,2,1^2], the two middle elements are 1 and 2; the low median is the smaller, 1.
First 8 rows:
1
1 1
2 0 1
3 1 0 1
4 2 0 0 1
6 3 1 0 0 1
8 4 2 0 0 0 1
11 6 3 1 0 0 0 1
Row n = 8 counts the following partitions:
(71) (62) (53) (44) . . . (8)
(611) (521) (431)
(5111) (422) (332)
(4211) (3221)
(41111) (2222)
(3311) (22211)
(32111)
(311111)
(221111)
(2111111)
(11111111)
(End)
MATHEMATICA
Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 2)/2]]] &, IntegerPartitions[#]]] &, Range[13]] (* Peter J. C. Moses, May 14 2019 *)
CROSSREFS
The high version of this triangle is A124944. The rank statistic for this triangle is A363941, high version A363942. A version for mean instead of median is A363945, rank statistic A363943. A high version for mean instead of median is A363946, rank stat A363944. A008284 counts partitions by length (or decreasing mean), strict A008289.
A360005(n)/2 returns median of prime indices.
Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.
+10
24
1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 280, 288, 300, 320, 324, 336, 352
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 30: {1,2,3}
2: {1} 32: {1,1,1,1,1}
3: {2} 36: {1,1,2,2}
4: {1,1} 40: {1,1,1,3}
6: {1,2} 48: {1,1,1,1,2}
8: {1,1,1} 54: {1,2,2,2}
9: {2,2} 56: {1,1,1,4}
10: {1,3} 60: {1,1,2,3}
12: {1,1,2} 64: {1,1,1,1,1,1}
16: {1,1,1,1} 72: {1,1,1,2,2}
18: {1,2,2} 80: {1,1,1,1,3}
20: {1,1,3} 81: {2,2,2,2}
24: {1,1,1,2} 84: {1,1,2,4}
27: {2,2,2} 88: {1,1,1,5}
28: {1,1,4} 90: {1,2,2,3}
MATHEMATICA
Select[Range[100], PrimeOmega[#]>=Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]/2&]
CROSSREFS
The case with difference at least 1 is A322136. A300061 lists numbers whose sum of prime indices is even.
Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.
Table, number of partitions of n with k as high median.
+10
23
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3
COMMENTS
For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017 Arrange the parts of a partition nonincreasing order. Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019
EXAMPLE
For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
Triangle begins:
1
1 1
1 1 1
2 1 1 1
3 1 1 1 1
4 3 1 1 1 1
6 4 1 1 1 1 1
8 6 3 1 1 1 1 1
11 8 5 1 1 1 1 1 1
15 11 7 3 1 1 1 1 1 1
20 15 9 5 1 1 1 1 1 1 1
26 21 12 8 3 1 1 1 1 1 1 1
35 27 16 10 5 1 1 1 1 1 1 1 1
45 37 21 13 8 3 1 1 1 1 1 1 1 1
58 48 29 16 11 5 1 1 1 1 1 1 1 1 1
Row n = 8 counts the following partitions:
(611) (521) (431) (44) (53) (62) (71) (8)
(5111) (422) (332)
(41111) (4211) (3311)
(32111) (3221)
(311111) (2222)
(221111) (22211)
(2111111)
(11111111)
(End)
MATHEMATICA
Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]] (* Peter J. C. Moses, May 14 2019 *)
CROSSREFS
The low version of this triangle is A124943. A008284 counts partitions by length, maximum, or decreasing mean.
A360005(n)/2 returns median of prime indices.
Search completed in 0.026 seconds
|