A003242 -id:A003242 - cp's OEIS frontend

A335433 Numbers whose multiset of prime indices is separable.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

First differs from A212167 in having 72.
Includes all squarefree numbers A005117.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
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.
Also Heinz numbers of separable partitions (A325534). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers that cannot be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.

			The sequence of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

The version for a multiset with prescribed multiplicities is A335127.
Separable factorizations are counted by A335434.
The complement is A335448.
Separations are counted by A003242 and A335452 and ranked by A333489.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535.
Strict permutations of prime indices are counted by A335489.
Cf. A000961, A005117, A056239, A112798, A114938, A181796, A292884, A335407, A335451, A335516.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{_,x_,x_,_}]&]!={}&]

A325535 Number of inseparable partitions of n; see Comments.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 16, 19, 28, 35, 48, 60, 79, 99, 131, 161, 205, 256, 324, 397, 498, 609, 755, 921, 1131, 1372, 1677, 2022, 2452, 2952, 3561, 4260, 5116, 6102, 7291, 8667, 10309, 12210, 14477, 17087, 20177, 23752, 27957, 32804, 38496, 45049, 52704

Offset: 0

Clark Kimberling, May 08 2019

Definition: a partition is separable if there is an ordering of its parts in which no consecutive parts are identical; otherwise the partition is inseparable.

A partition with k parts is inseparable if and only if there is a part whose multiplicity is greater than ceiling(k/2). - Andrew Howroyd, Jan 17 2024

			For n=5, the partition 1+2+2 is separable as 2+1+2, and 2+1+1+1 is inseparable.
From _Gus Wiseman_, Jun 27 2020: (Start)
The a(2) = 2 through a(9) = 11 inseparable partitions:
  11   111   22     2111    33       2221      44         333
             1111   11111   222      4111      2222       3222
                            3111     31111     5111       6111
                            21111    211111    41111      22221
                            111111   1111111   221111     51111
                                               311111     321111
                                               2111111    411111
                                               11111111   2211111
                                                          3111111
                                                          21111111
                                                          111111111
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are given by A335448.
Strict partitions are counted by A000009 and are all separable.
Anti-run compositions are counted by A003242.
Anti-run patterns are counted by A005649.
Partitions whose differences are an anti-run are A238424.
Separable partitions are counted by A325534.
Anti-run compositions are ranked by A333489.
Anti-run permutations of prime indices are counted by A335452.
Cf. A000041, A106356, A238594, A261962, A292884, A332668, A333175.

u=Table[Length[Select[Map[Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]] &,
IntegerPartitions[nn]], # > 1 &]], {nn, 50}]
Table[PartitionsP[n] - u[[n]], {n, 1, Length[u]}]
(* Peter J. C. Moses, May 07 2019 *)
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,10}] (* Gus Wiseman, Jun 27 2020 *)

seq(n) = {Vec(sum(k=1, (n+1)\2, x^(2*k-1)*(1 + x - x^(k-1))/((1-x^(k+1))*prod(j=1, k-1, 1 - x^j, 1 + O(x^(n-2*k+2)))), O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Jan 17 2024

a(n) = A000041(n) - A325534(n).

a(n) = Sum_{k>=1} x^(2*k-1)*(1 + x - x^(k-1))/((1-x^(k+1))*Product_{j=1..k-1} (1 - x^j)). - Andrew Howroyd, Jan 17 2024

a(0)=0 prepended by Andrew Howroyd, Jan 31 2024

A001250 Number of alternating permutations of order n.

Original entry on oeis.org

1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, 707584, 5405530, 44736512, 398721962, 3807514624, 38783024290, 419730685952, 4809759350882, 58177770225664, 740742376475050, 9902996106248192, 138697748786275802, 2030847773013704704, 31029068327114173810

Offset: 0

N. J. A. Sloane

For n>1, a(n) is the number of permutations of order n with the length of longest run equal 2.
Boustrophedon transform of the Euler numbers (A000111). [Berry et al., 2013] - N. J. A. Sloane, Nov 18 2013
Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i >= j < k or i < j >= k. a(4) = 10: 0010, 0011, 0020, 0021, 0022, 0101, 0102, 0103, 0112, 0113. - Alois P. Heinz, Oct 16 2019

			1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 32*x^5 + 122*x^6 + 544*x^7 + 2770*x^8 + ...
From _Gus Wiseman_, Jun 21 2021: (Start)
The a(0) = 1 through a(4) = 10 permutations:
  ()  (1)  (1,2)  (1,3,2)  (1,3,2,4)
           (2,1)  (2,1,3)  (1,4,2,3)
                  (2,3,1)  (2,1,4,3)
                  (3,1,2)  (2,3,1,4)
                           (2,4,1,3)
                           (3,1,4,2)
                           (3,2,4,1)
                           (3,4,1,2)
                           (4,1,3,2)
                           (4,2,3,1)
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..500 (terms n=1..100 from Max Alekseyev)
Max A. Alekseyev, On the number of permutations with bounded run lengths, arXiv:1205.4581 [math.CO], 2012-2013.
Désiré André, Sur les permutations alternées, J. Math. Pur. Appl., 7 (1881), 167-184.
Désiré André, Étude sur les maxima, minima et séquences des permutations, Ann. Sci. Ecole Norm. Sup., 3, no. 1 (1884), 121-135.
Désiré André, Mémoire sur les permutations quasi-alternées, Journal de mathématiques pures et appliquées 5e série, tome 1 (1895), 315-350.
Désiré André, Mémoire sur les séquences des permutations circulaires, Bulletin de la S. M. F., tome 23 (1895), pp. 122-184.
Stefano Barbero, Umberto Cerruti, and Nadir Murru, Some combinatorial properties of the Hurwitz series ring arXiv:1710.05665 [math.NT], 2017.
D. Berry, J. Broom, D. Dixon, and A. Flaherty, Umbral Calculus and the Boustrophedon Transform, 2013.
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
C. Davis, Problem 4755, Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534.
Chandler Davis, Problem 4755: A Permutation Problem, Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534. [Denoted by P_n in solution.] [Annotated scanned copy]
S. Kitaev, Multi-avoidance of generalized patterns, Discrete Math., 260 (2003), 89-100. (See p. 100.)
S. T. Thompson, Problem E754: Skew Ordered Sequences, Amer. Math. Monthly, 54 (1947), 416-417. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Alternating Permutation

Cf. A000111. A diagonal of A010094.
Cf. A001251, A001252, A001253, A010026, A211318, A260786.
The version for permutations of prime indices is A345164.
The version for compositions is A025047, ranked by A345167.
The version for patterns is A345194.
A049774 counts permutations avoiding adjacent (1,2,3).
A344614 counts compositions avoiding adjacent (1,2,3) and (3,2,1).
A344615 counts compositions avoiding the weak adjacent pattern (1,2,3).
A344654 counts partitions without a wiggly permutation, ranked by A344653.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions, ranked by A345168.
Cf. A000041, A003242, A032020, A056986, A261962, A325534, A325535, A335452, A344652, A344740, A345165.
Row sums of A104345.

a001250 n = if n == 1 then 1 else 2 * a000111 n
-- Reinhard Zumkeller, Sep 17 2014

# With Eulerian polynomials:
A := (n, x) -> `if`(n<2, 1/2/(1+I)^(1-n), add(add((-1)^j*binomial(n+1, j)*(m+1-j)^n, j=0..m)*x^m, m=0..n-1)):
A001250 := n -> 2*(I-1)^(1-n)*exp(I*(n-1)*Pi/2)*A(n,I);
seq(A001250(i), i=0..22); # Peter Luschny, May 27 2012
# second Maple program:
b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> `if`(n<2, 1, 2)*b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 29 2015

a[n_] := 4*Abs[PolyLog[-n, I]]; a[0] = a[1] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 09 2016, after M. F. Hasler *)
Table[Length[Select[Permutations[Range[n]],And@@(!(OrderedQ[#]||OrderedQ[Reverse[#]])&/@Partition[#,3,1])&]],{n,8}] (* Gus Wiseman, Jun 21 2021 *)
a[0]:=1; a[1]:=1; a[n_]:=a[n]=1/(n (n-1)) Sum[a[n-1-k] a[k] k, {k,1, n-1}]; Join[{a[0], a[1]}, Map[2 #! a[#]&, Range[2,24]]] (* Oliver Seipel, May 27 2024 *)

{a(n) = local(v=[1], t); if( n<0, 0, for( k=2, n+3, t=0; v = vector( k, i, if( i>1, t += v[k+1 - i]))); v[3])} /* Michael Somos, Feb 03 2004 */

{a(n) = if( n<0, 0, n! * polcoeff( (tan(x + x * O(x^n)) + 1 / cos(x + x * O(x^n)))^2, n))} /* Michael Somos, Feb 05 2011 */

A001250(n)=sum(m=0,n\2,my(k);(-1)^m*sum(j=0,k=n+1-2*m,binomial(k,j)*(-1)^j*(k-2*j)^(n+1))/k>>k)*2-(n==1)  \\ M. F. Hasler, May 19 2012

A001250(n)=4*abs(polylog(-n,I))-(n==1)  \\ M. F. Hasler, May 20 2012

my(x='x+O('x^66), egf=1+2*(tan(x)+1/cos(x))-2-x); Vec(serlaplace(egf)) /* Joerg Arndt, May 28 2012 */

from itertools import accumulate, islice
def A001250_gen(): # generator of terms
    yield from (1,1)
    blist = (0,2)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A001250_list = list(islice(A001250_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

from sympy import bernoulli, euler
def A001250(n): return 1 if n<2 else abs(((1<Chai Wah Wu, Nov 13 2024

# Algorithm of L. Seidel (1877)
def A001250_list(n) :
    R = [1]; A = {-1:0, 0:2}; k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        if i > 1 : R.append(A[-i//2] if i%2 == 0 else A[i//2])
    return R
A001250_list(22) # Peter Luschny, Mar 31 2012

a(n) = coefficient of x^(n-1)/(n-1)! in power series expansion of (tan(x) + sec(x))^2 = (tan(x)+1/cos(x))^2.
a(n) = coefficient of x^n/n! in power series expansion of 2*(tan(x) + sec(x)) - 2 - x. - Michael Somos, Feb 05 2011
For n>1, a(n) = 2 * A000111(n). - Michael Somos, Mar 19 2011
a(n) = 4*|Li_{-n}(i)| - [n=1] = Sum_{m=0..n/2} (-1)^m*2^(1-k)*Sum_{j=0..k} binomial(k,j)*(-1)^j*(k-2*j)^(n+1)/k - [n=1], where k = k(m) = n+1-2*m and [n=1] equals 1 if n=1 and zero else; Li denotes the polylogarithm (and i^2 = -1). - M. F. Hasler, May 20 2012
From Sergei N. Gladkovskii, Jun 18 2012: (Start)
Let E(x) = 2/(1-sin(x))-1 (essentially the e.g.f.), then
E(x) = -1 + 2*(-1/x + 1/(1-x)/x - x^3/((1-x)*((1-x)*G(0) + x^2))) where G(k) = (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction, Euler's 1st kind, 1-step).
E(x) = -1 + 2*(-1/x + 1/(1-x)/x - x^3/((1-x)*((1-x)*G(0) + x^2))) where G(k) = 8*k + 6 - x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction, Euler's 2nd kind, 2-step).
E(x) = (tan(x) + sec(x))^2 = -1 + 2/(1-x*G(0)) where G(k) = 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction, 3rd kind, 3-step).
(End)
G.f.: conjecture: 2*T(0)/(1-x) -1, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013
a(n) ~ 2^(n+3) * n! / Pi^(n+1). - Vaclav Kotesovec, Sep 06 2014
a(n) = Sum_{k=0..n-1} A109449(n-1,k)*A000111(k). - Reinhard Zumkeller, Sep 17 2014

Edited by Max Alekseyev, May 04 2012

a(0)=1 prepended by Alois P. Heinz, Nov 29 2015

A239455 Number of Look-and-Say partitions of n; see Comments.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 28, 33, 45, 55, 65, 83, 105, 121, 155, 180, 217, 259, 318, 362, 445, 512, 614, 707, 850, 958, 1155, 1309, 1543, 1754, 2079, 2327, 2740, 3085, 3592, 4042, 4699, 5253, 6093, 6815, 7839, 8751, 10069, 11208, 12832, 14266, 16270

Offset: 0

Clark Kimberling and Peter J. C. Moses, Mar 19 2014

nonn

Suppose that p = x(1) >= x(2) >= ... >= x(k) is a partition of n. Let y(1) > y(2) > ... > y(h) be the distinct parts of p, and let m(i) be the multiplicity of y(i) for 1 <= i <= h. Then we can "look" at p as "m(1) y(1)'s and m(2) y(2)'s and ... m(h) y(h)'s". Reversing the m's and y's, we can then "say" the Look-and-Say partition of p, denoted by LS(p). The name "Look-and-Say" follows the example of Look-and-Say integer sequences (e.g., A005150). As p ranges through the partitions of n, LS(p) ranges through all the Look-and-Say partitions of n. The number of these is A239455(n).
The Look-and-Say array is distinct from the Wilf array, described at A098859; for example, the number of Look-and-Say partitions of 9 is A239455(9) = 16, whereas the number of Wilf partitions of 9 is A098859(9) = 15. The Look-and-Say partition of 9 which is not a Wilf partition of 9 is [2,2,2,1,1,1].
Conjecture: a partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are all distinct, so y is counted under a(9). - Gus Wiseman, Aug 11 2025
Also the number of integer partitions y of n such that there is a pairwise disjoint way to choose a strict integer partition of each multiplicity (or run-length) of y. - Gus Wiseman, Aug 11 2025

			The 11 partitions of 6 generate 7 Look-and-Say partitions as follows:
6 -> 111111
51 -> 111111
42 -> 111111
411 -> 21111
33 -> 222
321 -> 111111
3111 -> 3111
222 -> 33
2211 -> 222
21111 -> 411
111111 -> 6,
so that a(6) counts these 7 partitions: 111111, 21111, 222, 3111, 33, 411, 6.

These include all Wilf partitions, counted by A098859, ranked by A130091.
These partitions are listed by A239454 in graded reverse-lex order.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351204, complement A351203.
The complement is counted by A351293, apparently ranked by A351295, conjugate A381433.
These partitions appear to be ranked by A351294, conjugate A381432.
The non-Wilf case is counted by A351592.
For normal multisets we appear to have A386580, complement A386581.
A000110 counts set partitions, ordered A000670.
A000569 = graphical partitions, complement A339617.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A181819 = Heinz number of the prime signature of n (prime shadow).
A279790 counts disjoint families on strongly normal multisets.
A329738 = compositions with all equal run-lengths.
A386583 counts separable partitions, sums A325534, ranks A335433.
A386584 counts inseparable partitions, sums A325535, ranks A335448.
A386585 counts separable type partitions, sums A336106, ranks A335127.
A386586 counts inseparable type partitions, sums A386638 or A025065, ranks A335126.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018, ranked by A044813.
- A329739 = compositions, for runs A351013, ranked by A351596.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A297770, A304660, A305563, A329746, A351201, A351202, A351291.

LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]]; TableForm[t = Map[LS[#] &, Range[10]]](*A239454,array*)
Flatten[t](*A239454,sequence*)
Map[Length[LS[#]] &, Range[25]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]>0&]],{n,0,10}] (* Gus Wiseman, Aug 11 2025 *)

A333381 Number of maximal anti-runs of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 3, 5, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 3, 2

Offset: 0

Gus Wiseman, Mar 24 2020

nonn

Anti-runs are sequences without any adjacent equal terms.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.
For n > 0, also one plus the number of adjacent equal pairs in the n-th composition in standard order.

			The 46th composition in standard order is (2,1,1,2), with maximal anti-runs ((2,1),(1,2)), so a(46) = 2.

Wikipedia, Longest increasing subsequence

Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are ranked by A333489.
Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],UnsameQ]],{n,0,100}]

For n > 0, a(n) = A124762(n) + 1.

A274174 Number of compositions of n if all summand runs are kept together.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 60, 97, 162, 254, 406, 628, 974, 1514, 2305, 3492, 5254, 7842, 11598, 17292, 25294, 37090, 53866, 78113, 112224, 161092, 230788, 328352, 466040, 658846, 928132, 1302290, 1821770, 2537156, 3536445, 4897310, 6777806, 9341456, 12858960, 17625970, 24133832, 32910898, 44813228, 60922160, 82569722

Offset: 0

Gregory L. Simay, Jun 12 2016

nonn

a(n^2) is odd. - Gregory L. Simay, Jun 23 2019

Also the number of compositions of n avoiding the patterns (1,2,1) and (2,1,2). - Gus Wiseman, Jul 07 2020

			If the summand runs are blocked together, there are 22 compositions of a(6): 6; 5+1, 1+5, 4+2, 2+4, (3+3), 4+(1+1), (1+1)+4, 1+2+3, 1+3+2, 2+1+3, 2+3+1, 3+1+2, 3+2+1, (2+2+2), 3+(1+1+1), (1+1+1)+3, (2+2)+(1+1), (1+1)+(2+2), 2+(1+1+1+1), (1+1+1+1)+2, (1+1+1+1+1+1).
a(0)=1; a(1)= 1; a(4) = 7; a(9) = 97; a(16) = 2305; a(25) = 78113 and a(36) = 3536445. - _Gregory L. Simay_, Jun 23 2019

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000070, A101880, A116608.
The version for patterns is A001339.
The version for prime indices is A333175.
The complement (i.e., the matching version) is A335548.
Anti-run compositions are A003242.
(1,2,1)- and (2,1,2)-matching permutations of prime indices are A335462.
(1,2,1)-matching compositions are A335470.
(1,2,1)-avoiding compositions are A335471.
(2,1,2)-matching compositions are A335472.
(2,1,2)-avoiding compositions are A335473.
Cf. A000670, A056986, A181796, A335451, A335452, A335460, A335463.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
       add(b(n-i*j, i-1, p+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 12 2016

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==Length[Union[#]]&]],{n,0,10}] (* Gus Wiseman, Jul 07 2020 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
    Sum[b[n - i*j, i - 1, p + If[j == 0, 0, 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 11 2021, after Alois P. Heinz *)

a(n) = Sum_{k>=0} k! * A116608(n,k). - Joerg Arndt, Jun 12 2016

Terms a(9) and beyond from Joerg Arndt, Jun 12 2016

A344612 Triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-alternating sum k ranging from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 3, 3, 1, 1, 0, 1, 2, 4, 3, 3, 1, 1, 0, 1, 2, 4, 5, 5, 3, 1, 1, 0, 1, 2, 4, 7, 5, 6, 3, 1, 1, 0, 1, 2, 4, 8, 7, 9, 6, 3, 1, 1, 0, 1, 2, 4, 8, 12, 7, 11, 6, 3, 1, 1, 0, 1, 2, 4, 8, 14, 11, 14, 12, 6, 3, 1, 1

Offset: 0

Gus Wiseman, Jun 01 2021

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is also (-1)^(k-1) times the sum of the even-indexed parts minus the sum of the odd-indexed parts.
Also the number of reversed integer partitions of n with alternating sum k ranging from -n to n in steps of 2.
Also the number of integer partitions of n with (-1)^(m-1) * b = k where m is the greatest part and b is the number of odd parts, with k ranging from -n to n in steps of 2.

			Triangle begins:
                                1
                              0   1
                            0   1   1
                          0   1   1   1
                        0   1   2   1   1
                      0   1   2   2   1   1
                    0   1   2   3   3   1   1
                  0   1   2   4   3   3   1   1
                0   1   2   4   5   5   3   1   1
              0   1   2   4   7   5   6   3   1   1
            0   1   2   4   8   7   9   6   3   1   1
          0   1   2   4   8  12   7  11   6   3   1   1
        0   1   2   4   8  14  11  14  12   6   3   1   1
      0   1   2   4   8  15  19  11  18  12   6   3   1   1
    0   1   2   4   8  15  24  15  23  20  12   6   3   1   1
  0   1   2   4   8  15  26  30  15  31  21  12   6   3   1   1
For example, row n = 7 counts the following partitions:
  (61)  (52)    (43)      (331)      (322)    (511)  (7)
        (4111)  (2221)    (22111)    (421)
                (3211)    (1111111)  (31111)
                (211111)
Row n = 9 counts the following partitions:
  81  72    63      54        441        333      522    711  9
      6111  4221    3222      22221      432      621
            5211    3321      33111      531      51111
            411111  4311      2211111    32211
                    222111    111111111  42111
                    321111               3111111
                    21111111

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041.
The midline k = n/2 is also A000041.
The right half (i.e., k >= 0) for even n is A344610.
The rows appear to converge to A344611 (from left) and A006330 (from right).
The non-reversed version is A344651 (A239830 interleaved with A239829).
The strict version is A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A003242, A027187, A124754, A152146, A344607, A344608, A344649, A344650, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]==k&]],{n,0,15},{k,-n,n,2}]

row(n)={my(v=vector(n+1)); forpart(p=n, my(s=-sum(i=1, #p, p[i]*(-1)^i)); v[(s+n)/2+1]++); v} \\ Andrew Howroyd, Jan 06 2024

A005649 Expansion of e.g.f. (2 - e^x)^(-2).

Original entry on oeis.org

1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548, 54667412, 862440068, 14857100084, 277474957988, 5584100659412, 120462266974148, 2772968936479604, 67843210855558628, 1757952715142990612, 48093560991292628228, 1385244691781856307124

Offset: 0

N. J. A. Sloane, Simon Plouffe

Exponential self-convolution of numbers of preferential arrangements.
Number of compatible bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003
Stirling transform of A000142, shifted left one place: 1, 2, 6, 24, 120, 720, ... - Philippe Deléham, May 17 2005; corrected by Ilya Gutkovskiy, Jul 25 2018
With an extra 1 at the beginning, coefficients of the formal (divergent) series expansion at infinity of Sum_{k>=0} 1/binomial(x,k) = 1+1/x+2/x^2+8/x^3+... Also Sum_{k>=0} k!/x^k Product_{i=1..k-1} 1/(1-i/x) yields a generating function in 1/x. - Roland Bacher, Nov 21 2000
Stirling-Bernoulli transform of A001057: 1, -1, 2, -2, 3, -3, 4, ... - Philippe Deléham, May 27 2015
a(n) is the total number of open sets summed over all chain topologies that can be placed on an n-set.  A chain topology is a topology whose open sets can be totally ordered by inclusion. - Geoffrey Critzer, Apr 06 2017
From Gus Wiseman, Jun 10 2020: (Start)
Also the number of length n + 1 sequences covering an initial interval of positive integers with no adjacent equal parts (anti-runs). For example, the a(0) = 1 through a(2) = 8 anti-runs are:
  (1)  (1,2)  (1,2,1)
       (2,1)  (1,2,3)
              (1,3,2)
              (2,1,2)
              (2,1,3)
              (2,3,1)
              (3,1,2)
              (3,2,1)
Also the number of ordered set partitions of {1,...,n + 1} with no two successive vertices in the same block. For example, the a(0) = 1 through a(2) = 8 ordered set partitions are:
  {{1}}  {{1},{2}}   {{1,3},{2}}
         {{2},{1}}   {{2},{1,3}}
                    {{1},{2},{3}}
                    {{1},{3},{2}}
                    {{2},{1},{3}}
                    {{2},{3},{1}}
                    {{3},{1},{2}}
                    {{3},{2},{1}}
(End)
From Manfred Boergens, Feb 24 2025: (Start)
a(n+1) is the n-th row sum in A380977.
Number of surjections f with domain [n+1] and f(n+1)!=f(j) for jNumber of (n+1)-tuples containing all elements of a set, with a unique last element.
Consider an urn with balls of pairwise different colors. a(n) is the number of (n+1)-sequences of draws with replacement completing the covering of all colors with the last draw, the number of colors running from 1 to n+1.
(End)

			a(2)=8 gives the number of 3-tuples containing all elements of a set [n] with n<=3 and a unique last element: 112, 221, 123, 213, 132, 312, 231, 321. - _Manfred Boergens_, Feb 24 2025

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..423 (first 101 terms from T. D. Noe)
José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, Generalized Barred Preferential Arrangements, Transactions on Combinatorics (2022).
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
D. Foata, and C. Krattenthaler, Graphical Major Indices, II, Séminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 154
Augustine O. Munagi, Two Applications of the Bijection on Fibonacci Set Partitions, Fibonacci Quart. 55 (2017), no. 5, 144-148.

Cf. A000670.
2*A083410(n)=a(n), if n>0.
Pairwise sums of A052841 and also of A089677.
Anti-run compositions are counted by A003242.
A triangle counting maximal anti-runs of compositions is A106356.
Anti-runs of standard compositions are counted by A333381.
Adjacent unequal pairs in standard compositions are counted by A333382.
Cf. A000110, A008277, A055932, A124762, A238279, A238424, A317081.
Cf. A380977.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2021

f[n_] := Sum[(i + j)^n/2^(2 + i + j), {i, 0, Infinity}, {j, 0, Infinity}]; Array[f, 20, 0] (* Vladimir Reshetnikov, Dec 31 2008 *)
a[n_] := (-1)^n (PolyLog[-n-1, 2] - PolyLog[-n, 2])/4; Array[f, 20, 0] (* Vladimir Reshetnikov, Jan 23 2011 *)
Range[0, 19]! CoefficientList[Series[(2 - Exp@ x)^-2, {x, 0, 19}], x] (* Robert G. Wilson v, Jan 23 2011 *)
nn = 19; Range[0, nn]! CoefficientList[Series[1 + D[u^2 (Exp[z] - 1)/(1 - u (Exp[z] - 1)), u] /. u -> 1, {z, 0, nn}], z] (* Geoffrey Critzer, Apr 06 2017 *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],FreeQ[Differences[#],0]&]],{n,0,6}] (* Gus Wiseman, Jun 10 2020 *)
With[{nn=20},CoefficientList[Series[1/(2-E^x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 02 2021 *)
Table[Sum[(m+1)! StirlingS2[n,m],{m,0,n}],{n,0,19}] (* Manfred Boergens, Feb 24 2025 *)

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(binomial(n,k)*t(k)*t(n-k),k,0,n),n,0,20);
/* Emanuele Munarini, Oct 02 2012 */

a(n)=if(n<0,0,n!*polcoeff(subst(1/(1-y)^2,y,exp(x+x*O(x^n))-1),n))

a(n)=polcoeff(sum(m=0, n,(2*m)!/m!*x^m/prod(k=1, m,1+(m+k)*x+x*O(x^n))), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 03 2013

E.g.f.: 1/(2-exp(x))^2.
a(n) = (A000670(n) + A000670(n+1)) / 2. - Philippe Deléham, May 16 2005
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A052841. - Peter Bala, Nov 25 2011
E.g.f.: 1/(2-exp(x))^2 = 1/(G(0) + 4), G(k) = 1-4/((2^k)-x*(4^k)/((2^k)*x-(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
O.g.f.: Sum_{n>=0} (2*n)!/n! * x^n / Product_{k=1..n} (1 + (n+k)*x). - Paul D. Hanna, Jan 03 2013
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (k+1)/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1/G(0) where G(k) = 1 - x*(k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
a(n) = Sum_{k = 0..n} A163626(n,k) * A001057(k+1). - Philippe Deléham, May 27 2015
a(n) ~ n! * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=0..n} Stirling2(n,k)*(k + 1)!. - Ilya Gutkovskiy, Jul 25 2018
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A374629 Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 3, 3, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, Jul 20 2024

The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).
The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:
    0:      () -> ()      15: (1,1,1,1) -> (1)
    1:     (1) -> (1)     16:       (5) -> (5)
    2:     (2) -> (2)     17:     (4,1) -> (4,1)
    3:   (1,1) -> (1)     18:     (3,2) -> (3,2)
    4:     (3) -> (3)     19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2,1)   20:     (2,3) -> (2)
    6:   (1,2) -> (1)     21:   (2,2,1) -> (2,1)
    7: (1,1,1) -> (1)     22:   (2,1,2) -> (2,1)
    8:     (4) -> (4)     23: (2,1,1,1) -> (2,1)
    9:   (3,1) -> (3,1)   24:     (1,4) -> (1)
   10:   (2,2) -> (2)     25:   (1,3,1) -> (1,1)
   11: (2,1,1) -> (2,1)   26:   (1,2,2) -> (1)
   12:   (1,3) -> (1)     27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1,1)   28:   (1,1,3) -> (1)
   14: (1,1,2) -> (1)     29: (1,1,2,1) -> (1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-leaders are A065120.
Row-lengths are A124766.
Row-sums are A374630.
Positions of constant rows are A374633, counted by A374631.
Positions of strict rows are A374768, counted by A374632.
For other types of runs we have A374251, A374515, A374683, A374740, A374757.
Positions of non-weakly decreasing rows are A375137.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627, length A124767, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A046660, A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637, A374701, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[First/@Split[stc[n],LessEqual],{n,0,100}]

Previous Showing 11-20 of 507 results. Next

cp's OEIS Frontend

A335448 Numbers whose prime indices are inseparable.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A335433 Numbers whose multiset of prime indices is separable.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325535 Number of inseparable partitions of n; see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A001250 Number of alternating permutations of order n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Python

Python

Sage

Formula

Extensions

A239455 Number of Look-and-Say partitions of n; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A333381 Number of maximal anti-runs of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A274174 Number of compositions of n if all summand runs are kept together.

Views

Author

Keywords

Comments