A181796 -id:A181796 - cp's OEIS frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

This sequence does not contain any number of the form 36n-6 or 36n+6, as such numbers are divisible by 6 but not by 4 or 9. Consequently, this sequence does not contain 24 consecutive integers. The quest for the greatest number of consecutive integers in this sequence has ties to the ABC conjecture (see the MathOverflow link). - Danny Rorabaugh, Sep 23 2015
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities. The enumeration of these partitions by sum is given by A098859. - Gus Wiseman, May 04 2019
Aktaş and Ram Murty (2017) called these terms "special numbers" ("for lack of a better word"). They prove that the number of terms below x is ~ c*x/log(x), where c > 1 is a constant. - Amiram Eldar, Feb 25 2021
Sequence A005940(1+A328592(n)), n >= 1, sorted into ascending order. - Antti Karttunen, Apr 03 2022

			From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
MathOverflow, Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, arXiv:1902.09224 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Factorization

Complement of A130092. A351564 is the characteristic function.
Subsequence of A351294.
Cf. A000961, A006939, A181818, A304686, A319161, A342028, A342029, A342030, A342031, A342032 (subsequences).
Cf. A005940, A048767, A048768, A056239, A098859, A112798, A118914, A181796, A217605, A325326, A325337, A325368, A327498, A327523, A328592, A336423, A336424, A336569, A336570, A336571, A343012, A343013.

filter:= proc(t) local f;
f:= map2(op,2,ifactors(t)[2]);
nops(f) = nops(convert(f,set));
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 30 2015

t[n_] := FactorInteger[n][[All, 2]]; Select[Range[400],  Union[t[#]] == Sort[t[#]] &]  (* Clark Kimberling, Mar 12 2015 *)

isok(n) = {nbf = omega(n); f = factor(n); for (i = 1, nbf, for (j = i+1, nbf, if (f[i, 2] == f[j, 2], return (0)););); return (1);} \\ Michel Marcus, Aug 18 2013

isA130091(n) = issquarefree(factorback(apply(e->prime(e), (factor(n)[, 2])))); \\ Antti Karttunen, Apr 03 2022

a(n) < A130092(n) for n<=150, a(n) > A130092(n) for n>150.

A032020 Number of compositions (ordered partitions) of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 11, 13, 19, 27, 57, 65, 101, 133, 193, 351, 435, 617, 851, 1177, 1555, 2751, 3297, 4757, 6293, 8761, 11305, 15603, 24315, 30461, 41867, 55741, 74875, 98043, 130809, 168425, 257405, 315973, 431065, 558327, 751491, 958265, 1277867, 1621273

Offset: 0

Christian G. Bower, Apr 01 1998

Compositions into distinct parts are equivalent to (1,1)-avoiding compositions. - Gus Wiseman, Jun 25 2020

All terms are odd. - Alois P. Heinz, Apr 09 2021

			a(6) = 11 because 6 = 5+1 = 4+2 = 3+2+1 = 3+1+2 = 2+4 = 2+3+1 = 2+1+3 = 1+5 = 1+3+2 = 1+2+3.
From _Gus Wiseman_, Jun 25 2020: (Start)
The a(0) = 1 through a(7) = 13 strict compositions:
  ()  (1)  (2)  (3)    (4)    (5)    (6)      (7)
                (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
                (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                              (3,2)  (4,2)    (3,4)
                              (4,1)  (5,1)    (4,3)
                                     (1,2,3)  (5,2)
                                     (1,3,2)  (6,1)
                                     (2,1,3)  (1,2,4)
                                     (2,3,1)  (1,4,2)
                                     (3,1,2)  (2,1,4)
                                     (3,2,1)  (2,4,1)
                                              (4,1,2)
                                              (4,2,1)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
C. G. Bower, Transforms (2)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97. (free access)
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A008289, A032011.
Row sums of A241719.
Column k=1 of A243081, A242447, A261835, A261836, A327244, A327245.
Main diagonal of A261960.
Dominated by A003242 (anti-run compositions).
These compositions are ranked by A233564.
(1,1)-avoiding patterns are counted by A000142.
Numbers with strict prime signature are A130091.
(1,1,1)-avoiding compositions are counted by A232432.
(1,1)-matching compositions are counted by A261982.
Inseparable partitions are counted by A325535.
Patterns matched by compositions are counted by A335456.
Strict permutations of prime indices are counted by A335489.
Cf. A000009, A011782, A102726, A181796, A261962, A333489, A335457.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0))) end:
a:= proc(n) local l; l:=b(n, n): add((i-1)! *l[i], i=1..nops(l)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 12 2012
# second Maple program:
T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
    end:
a:= n-> add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 04 2015

f[list_]:=Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 0,30}]
T[n_, k_] := T[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], T[n-k, k] + k*T[n-k, k-1]]]; a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[8*n + 1] - 1) / 2]}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

N=66;  q='q+O('q^N);
gf=sum(n=0,N, n!*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf)
/* Joerg Arndt, Oct 20 2012 */

Q(N) = { \\ A008289
  my(q = vector(N)); q[1] = [1, 0, 0, 0];
  for (n = 2, N,
    my(m = (sqrtint(8*n+1) - 1)\2);
    q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
    for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
  return(q);
};
seq(N) = concat(1, apply(q -> sum(k = 1, #q, q[k] * k!), Q(N)));
seq(43) \\ Gheorghe Coserea, Sep 09 2018

"AGK" (ordered, elements, unlabeled) transform of 1, 1, 1, 1, ...
G.f.: Sum_{k>=0} k! * x^((k^2+k)/2) / Product_{j=1..k} (1-x^j). - David W. Wilson May 04 2000
a(n) = Sum_{m=1..n} A008289(n,m)*m!. - Geoffrey Critzer, Sep 07 2012

A335448 Numbers whose prime indices are inseparable.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

First differs from A212164 in lacking 72.
First differs from A293243 in lacking 72.
No terms are squarefree.
Also Heinz numbers of inseparable partitions (A325535). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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 also numbers that can be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.
A multiset is inseparable iff its maximal multiplicity is greater than one plus the sum of its remaining multiplicities.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
  32: {1,1,1,1,1}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  80: {1,1,1,1,3}
  81: {2,2,2,2}
  88: {1,1,1,5}
  96: {1,1,1,1,1,2}

Complement of A335433.
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. A000670, A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451.

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

A335433 Numbers whose multiset of prime indices is separable.

Original entry on oeis.org

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_,_}]&]!={}&]

A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1, 1, 7, 27, 76, 174, 343, 602, 961, 1415, 1940, 2493, 3017, 3450, 3736, 3836, 3736, 3450, 3017, 2493, 1940, 1415, 961, 602, 343, 174, 76, 27, 7, 1, 1, 8, 35, 111, 285, 628, 1230, 2191, 3606, 5545, 8031, 11021, 14395, 17957, 21450, 24584, 27073, 28675, 29228, 28675, 27073, 24584, 21450, 17957, 14395, 11021, 8031, 5545, 3606, 2191, 1230, 628, 285, 111, 35, 8, 1

Offset: 1

N. J. A. Sloane

T(n,k) is the number of permutations of {1..n} with k inversions.
n-th row gives growth series for symmetric group S_n with respect to transpositions (1,2), (2,3), ..., (n-1,n).
T(n,k) is the number of permutations of (1,2,...,n) having disorder equal to k. The disorder of a permutation p of (1,2,...,n) is defined in the following manner. We scan p from left to right as often as necessary until all its elements are removed in increasing order, scoring one point for each occasion on which an element is passed over and not removed. The disorder of p is the number of points scored by the end of the scanning and removal process. For example, the disorder of (3,5,2,1,4) is 8, since on the first scan, 3,5,2 and 4 are passed over, on the second, 3,5 and 4 and on the third scan, 5 is once again not removed. - Emeric Deutsch, Jun 09 2004
T(n,k) is the number of permutations p=(p(1),...,p(n)) of {1..n} such that Sum_{i: p(i)>p(i+1)} = k (k is called the Major index of p). Example: T(3,0)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1 because the major indices of the permutations (1,2,3), (2,1,3), (3,1,2), (1,3,2), (2,3,1) and (3,2,1) are 0,1,1,2,2 and 3, respectively. - Emeric Deutsch, Aug 17 2004
T(n,k) is the number of 2 X c matrices with column totals 1,2,3,...,n and row totals k and binomial(n+1,2) - k. - Mitch Harris, Jan 13 2006
T(n,k) is the number of permutations p of {1,2,...,n} for which den(p)=k. Here den is the Denert statistic, defined in the following way: let p=p(1)p(2)...p(n) be a permutation of {1,2,...,n}; if p(i)>i, then we say that i is an excedance of p; let i_1 < i_2 < ... < i_k be the excedances of p and let j_1 < j_2 < ... < j_{n-k} be the non-excedances of p; let Exc(p) = p(i_1)p(i_2)...p(i_k), Nexc(p)=p(j_1)p(j_2)...p(j_{n-k}); then, by definition den(p) = i_1 + i_2 + ... + i_k + inv(Exc(p)) + inv(Nexc(p)), where inv denotes "number of inversions". Example: T(4,5)=3 because we have 1342, 3241 and 4321. We show that den(4321)=5: the excedances are 1 and 2; Exc(4321)=43, Nexc(4321)=21; now den(4321) = 1 + 2 + inv(43) + inv(21) = 3+1+1 = 5. - Emeric Deutsch, Oct 29 2008
T(n,k) is the number of size k submultisets of the multiset {1,2,2,3,3,3,...,n-1} (which contains i copies of i for 0 < i < n).
The limit of products of the numbers of fixed necklaces of length n composed of beads of types N(n,b), n --> infinity, is the generating function for inversions (we must exclude one unimportant factor b^n/n!). The error is < (b^n/n!)*O(1/n^(1/2-epsilon)). See Gaichenkov link. - Mikhail Gaichenkov, Aug 27 2012
The number of ways to distribute k-1 indistinguishable balls into n-1 boxes of capacity 1,2,3,...,n-1. - Andrew Woods, Sep 26 2012
Partial sums of rows give triangle A161169. - András Salamon, Feb 16 2013
The number of permutations of n that require k pair swaps in the bubble sort to sort them into the natural 1,2,...,n order. - R. J. Mathar, May 04 2013
Also series coefficients of q-factorial [n]q ! -- see Mathematica line. - _Wouter Meeussen, Jul 12 2014
From Mikhail Gaichenkov, Aug 16 2016: (Start)
Following asymptotic expansions in the Central Limit Theorem developed by Valentin V. Petrov, the cumulative distribution function of these numbers, CDF_N(x), is equal to the CDF of the normal distribution - (0.06/sqrt(2*Pi))*exp(-x^2/2)(x^3-3x)*(6N^3+21N^2+31N+31)/(N(2N+5)^2(N-1)+O(1/N^2).
This can be written as: CDF of the normal distribution -(0.09/(N*sqrt(2*Pi)))*exp(-x^2/2)*He_3(x) + O(1/N^2), N > 1, natural numbers (Gaichenkov, private research).
According to B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4, "the unimodal behavior of the inversion numbers suggests that the number of inversions in a random permutation may be asymptotically normal". See links.
Moreover, E. Ben-Naim (Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory), "On the Mixing of Diffusing Particles" (13 Oct 2010), states that the Mahonian Distribution becomes a function of a single variable for large numbers of element, i.e., the probability distribution function is normal. See links.
To be more precise the expansion of the distribution is presented for a finite number of elements (or particles in terms of E. Ben-Naim's article). The distribution tends to the normal distribution for an infinite numbers of elements.
(End)
T(n,k) statistic counts (labeled) permutation graphs with n vertices and k edges. - Mikhail Gaichenkov, Aug 20 2019
From Gus Wiseman, Aug 12 2020: (Start)
Number of divisors of A006939(n - 1) or A076954(n - 1) with k prime factors, counted with multiplicity, where A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). For example, row n = 4 counts the following divisors:
  1   2   4   8  24  72 360
      3   6  12  36 120
      5   9  18  40 180
         10  20  60
         15  30  90
             45
Crossrefs:
A336420 is the case with distinct prime multiplicities.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A317829 counts factorizations of superprimorials.
A336941 counts divisor chains under superprimorials.
Cf. A000005, A001222, A008278, A022559, A027423, A076954, A181796.
(End)
Named after the British mathematician Percy Alexander MacMahon (1854-1929). - Amiram Eldar, Jun 13 2021
Row maxima ~ n!/(sigma * sqrt(2*Pi)), sigma^2 = (2*n^3 + 9*n^2 + 7*n)/72 = variance of group type A_n (see also A161435). - Mikhail Gaichenkov, Feb 08 2023
Sum_{i>=0} T(n,i)*k^i = A069777(n,k). - Geoffrey Critzer, Feb 26 2025

			1; 1+x; (1+x)*(1+x+x^2) = 1+2*x+2*x^2+x^3; etc.
Triangle begins:
  n\k| 0  1   2    3    4     5     6     7     8      9     10
  ---+--------------------------------------------------------------
   1 | 1;
   2 | 1, 1;
   3 | 1, 2,  2,   1;
   4 | 1, 3,  5,   6,   5,    3,    1;
   5 | 1, 4,  9,  15,  20,   22,   20,   15,    9,     4,     1;
   6 | 1, 5, 14,  29,  49,   71,   90,  101,  101,    90,    71, ...
   7 | 1, 6, 20,  49,  98,  169,  259,  359,  455,   531,   573, ...
   8 | 1, 7, 27,  76, 174,  343,  602,  961, 1415,  1940,  2493, ...
   9 | 1, 8, 35, 111, 285,  628, 1230, 2191, 3606,  5545,  8031, ...
  10 | 1, 9, 44, 155, 440, 1068, 2298, 4489, 8095, 13640, 21670, ...
From _Gus Wiseman_, Aug 12 2020: (Start)
Row n = 4 counts the following submultisets of {1,1,1,2,2,3}:
  {}  {1}  {11}  {111}  {1112}  {11122}  {111223}
      {2}  {12}  {112}  {1122}  {11123}
      {3}  {22}  {122}  {1113}  {11223}
           {13}  {113}  {1123}
           {23}  {123}  {1223}
                 {223}
(End)

Miklós Bóna, Combinatorics of permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004 (p. 52).
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 240.
Florence Nightingale David, Maurice George Kendall, and David Elliot Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 163, top display.
Eugen Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
Valentin V. Petrov, Sums of Independent Random Variables, Springer Berlin Heidelberg, 1975, p. 134.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.

Alois P. Heinz, Rows n = 1..50, flattened (first 30 rows from Paul D. Hanna)
Ashleigh Adams, Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, Cyclic sieving on permutations - an analysis of maps and statistics in the FindStat database, arXiv:2402.16251 [math.CO], 2024. See p. 7.
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics, Vol. 35, No. 1 (2019), pp. 1-11.
Hasan Arslan, A combinatorial interpretation of Mahonian numbers of type B, arXiv:2404.05099 [math.CO], 2024. See p. 3.
E. Ben-Naim, On the Mixing of Diffusing Particles, arXiv:1010.2563 [cond-mat.stat-mech], 2010.
David J. Benson, The cohomology of the nilCoxeter algebra, arXiv:2407.21175 [math.RA], 2024. See p. 4.
Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Tableaux posets and the fake degrees of coinvariant algebras, arXiv:1809.07386 [math.CO], 2018.
Niclas Boehmer, Robert Bredereck, Piotr Faliszewski, and Rolf Niedermeier, On the Robustness of Winners: Counting Briberies in Elections, arXiv:2010.09678 [cs.GT], 2020.
Niclas Boehmer, Robert Bredereck, Piotr Faliszewski, Rolf Niedermeier, and Stanisław Szufa, Putting a Compass on the Map of Elections, arXiv:2105.07815 [cs.GT], 2021.
J. Bourget, Des permutations, Nouvelles annales de mathématiques, 2e série, tome 10 (1871), pp. 254-268.
Franc Brglez, Of n-dimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, Elektrotehniski Vestnik, Vol. 78, No. 4 (2011), pp. 181-192.
Agnieszka Bukietyńska, The test of inversion in the analysis of investment funds, Central and Eastern European Journal of Management and Economics, Vol. 5, No. 3 (September 2017), pp. 277-289.
L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J., Vol. 15, No. 4 (1948), pp. 987-1000.
Luke Chamandy, Anvar Shukurov, and A. Russ Taylor, Statistical tests of galactic dynamo theory, arXiv:1609.05688 [astro-ph.GA], 2016.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Mariusz Czekała and Agnieszka Bukietyńska, Distribution of Inversions and the Power of the τ-Kendall's Test, in J. Świątek, Z. Wilimowska, L. Borzemski, A. Grzech (eds.), Information Systems Architecture and Technology: Proceedings of 37th International Conference on Information Systems Architecture and Technology - ISAT 2016 - Part III, pp. 175-185.
Mariusz Czekała, Agnieszka Bukietyńska, and Agnieszka Matylda Schlichtinger, Estimation of the Probability of Inversion in the Test of Variable Dependencies, Information Systems Architecture and Technology: Proceedings of 39th International Conference on Information Systems Architecture and Technology (ISAT 2018), Part III, 145-156.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, pp. 241-242. (Annotated scanned copy)
Emeric Deutsch, Problem 10975: Enumeration of Permutations by Disorder, Amer. Math. Monthly, Vol. 111, No. 6 (2004), p. 541.
FindStat - Combinatorial Statistic Finder, The Denert index of a permutation, The major index of a permutation, The number of inversions of a permutation, The sorting index of a permutation.
Dominique Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, in: M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977, pp. 27-49.
Dominique Foata and Doron Zeilberger, Denert's permutation statistic is indeed Euler-Mahonian, Studies in Appl. Math., Vol. 83 (1990), pp. 31-59. [From _Emeric Deutsch_, Oct 29 2008]
Víctor Franco-Sánchez, Arnau Martí-Llobet, and Ramon Ferrer-i-Cancho, Swap distance minimization beyond entropy minimization in word order variation, arXiv:2404.14192 [cs.CL], 2024. See p. 34.
Mikhail Gaichenkov, Necklaces and the generating function for inversions, MathOverflow, 2012.
Mikhail Gaichenkov, Pros and cons of probability model for permutations, MathOverflow, 2014.
Amy Grady, Sorting index and Mahonian-Stirling pairs for labeled forests, Clemson University, July 10, 2014.
Oğuz Gürerk, Ümit Işlak, and Mehmet Akif Yıldız, A study on random permutation graphs, arXiv:1901.06678 [math.CO], 2019.
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988.
Guo-Niu Han, Une nouvelle bijection pour la statistique de Denert, C. R. Acad. Sci. Paris, Ser. I, Vol. 310 (1990), pp. 493-496.
Stuart A. Hannah, Sieved Enumeration of Interval Orders and Other Fishburn Structures, arXiv:1502.05340 [math.CO], (18-February-2015).
Yang-Hui He, Cyril Matti and Chuang Sun, The Scattering Variety, arXiv preprint arXiv:1403.6833 [hep-th], 2014.
Ekhine Irurozki, Sampling and learning distance-based probability models for permutation spaces, PhD Dissertation, Department of Computer Science and Artificial Intelligence of the University of the Basque Country, 2015.
Ekhine Irurozki, Borja Calvo and José A. Lozano, An R package for permutations, Mallows and Generalized Mallows models, 2014.
Ekhine Irurozki, Borja Calvo and José A. Lozano, PerMallows: An R Package for Mallows and Generalized Mallows Models, Journal of Statistical Software, Vol. 71, No. 12 (August 2016).
Milan Janjic, A Generating Function for Numbers of Insets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.7.
Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.
James A. Koziol, Sums of ranking differences and inversion numbers for method discrimination, Journal of Chemometrics, Vol. 27 (2013), pp. 165-169.
Barbara H. Margolius, Permutations with inversions, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.
Jeremy L. Martin and Jennifer D. Wagner, On the Spectra of Simplicial Rook Graphs, arXiv preprint arXiv:1209.3493 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 27 2012
Anthony Mendes, A note on alternating permutations, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 437-440.
Vinicus de Morais Alves, Rafael Dowsley, Rafael Timateo de Sousa, and Anderson C.A. Nascimento, Information-Theoretically Secure String Commitments Based on Packet Reordering Channels, IEEE Access, Volume 9 (2021), pp. 139931-139936.
Roger H. Moritz and Robert C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., Vol. 61, No. 1 (1988), pp. 24-29.
Eugen Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.
Eugen Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.
Juan Miguel Nieto, Tailoring and Hexagon Form Factors, Spinning Strings and Correlation Functions in the AdS/CFT Correspondence, Springer Theses (Recognizing Outstanding Ph.D. Research), Springer, Cham, 2018.
Michal Opler, Major index distribution over permutation classes, arXiv:1505.07135 [math.CO], 2015.
Oren Patashnik, Letter to R. H. Moritz & R. C. Williams, cc N. J. A. Sloane, Oct 1988.
Michael Penn, The Quantum Factorial is upon us., YouTube video, 2023.
Nikolay L. Poliakov, Note on level r consensus, arXiv preprint arXiv:1606.04816 [q-fin.EC], 2016.
Svetlana Poznanovic, The sorting index and equidistribution of set-valued statistics over restricted permutations, Journal of Combinatorial Theory, Series A, Vol. 125 (2014), pp. 254-272.
Richard P. Stanley, Binomial posets, Moebius inversion and permutation enumeration, J. Combinat. Theory, Series A, Vol. 20, No. 3 (1976), pp. 336-356.
A. Waksman, On the complexity of inversions, IEEE Trans. Computers, Vol. 19 (1970), pp. 1225-1226. See Table II.
Eric Weisstein's World of Mathematics, Necklace.
Eric Weisstein's World of Mathematics, Irreducible Polynomial.
Thomas Wieder, Comments on A008302.
Wikipedia, Major index

Diagonals: A000707 (k=n-1), A001892 (k=n-2), A001893 (k=n-3), A001894 (k=n-4), A005283 (k=n-5), A005284 (k=n-6), A005285 (k=n-7).
Columns: A005286 (k=3), A005287 (k=4), A005288 (k=5), A242656 (k=6), A242657 (k=7).
Rows: A161435 (n=4), A161436 (n=5), A161437 (n=6), A161438 (n=7), A161439 (n=8), A161456 (n=9), A161457 (n=10).
Row-maxima: A000140, truncated table: A060701, row sums: A000142, row lengths: A000124.
A001809 gives total Denert index of all permutations.
A357611 gives a refinement.
Cf. A000217, A139365, A069777.

g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001
BB:=j->1+sum(t^i, i=1..j): for n from 1 to 8 do Z[n]:=sort(expand(simplify(product(BB(j), j=0..n-2)))) od: for n from 1 to 8 do seq(coeff(Z[n], t, j), j=0..(n-1)*(n-2)/2) od; # Zerinvary Lajos, Apr 13 2007
# alternative Maple program:
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+
       add(b(u-j, o+j-1)*x^(u-j), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 02 2017

f[n_] := CoefficientList[ Expand@ Product[ Sum[x^i, {i, 0, j}], {j, n}], x]; Flatten[Array[f, 8, 0]]
(* Second program: *)
T[0, 0] := 1; T[-1, k_] := 0;
T[n_, k_] := T[n, k] = If[0 <= k <= n*(n - 1)/2, T[n, k - 1] + T[n - 1, k] - T[n - 1, k - n], 0]; (* Peter Kagey, Mar 18 2021; corrected the program by Mats Granvik and Roger L. Bagula, Jun 19 2011 *)
alternatively (versions 7 and up):
Table[CoefficientList[Series[QFactorial[n,q],{q,0,n(n-1)/2}],q],{n,9}] (* Wouter Meeussen, Jul 12 2014 *)
b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
   Sum[b[u + j - 1, o - j]*x^(u + j - 1), {j, 1, o}] +
   Sum[b[u - j, o + j - 1]*x^(u - j), {j, 1, u}]]];
T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 21 2025, after Alois P. Heinz *)

{T(n,k) = my(A=1+x); for(i=1,n, A = 1 + intformal(A - q*subst(A,x,q*x +x^2*O(x^n)))/(1-q)); polcoeff(n!*polcoeff(A,n,x),k,q)}
for(n=1,10, for(k=0,n*(n-1)/2, print1(T(n,k),", ")); print("")) \\ Paul D. Hanna, Dec 31 2016

row(n)=Vec(prod(k=1,n,(1-'q^k)/(1-'q))); \\ Joerg Arndt, Apr 13 2019

from sage.combinat.q_analogues import q_factorial
for n in (1..6): print(q_factorial(n).list()) # Peter Luschny, Jul 18 2016

Bourget, Comtet and Moritz-Williams give recurrences.
Mendes and Stanley give g.f.'s.
G.f.: Product_{j=1..n} (1-x^j)/(1-x) = Sum_{k=0..M} T{n, k} x^k, where M = n*(n-1)/2.
From Andrew Woods, Sep 26 2012, corrected by Peter Kagey, Mar 18 2021: (Start)
T(1, 0) = 1,
T(n, k) = 0 for n < 0, k < 0 or k > n*(n-1)/2.
T(n, k) = Sum_{j=0..n-1} T(n-1, k-j),
T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-n). (End)
E.g.f. satisfies: A(x,q) = 1 + Integral (A(x,q) - q*A(q*x,q))/(1-q) dx, where A(x,q) = Sum_{n>=0} x^n/n! * Sum_{k=0..n*(n-1)/2} T(n,k)*q^k, when T(0,0) = 1 is included. - Paul D. Hanna, Dec 31 2016

There were some mistaken edits to this entry (inclusion of an initial 1, etc.) which I undid. - N. J. A. Sloane, Nov 30 2009

Added mention of "major index" to definition. - N. J. A. Sloane, Feb 10 2019

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

Original entry on oeis.org

1, 2, 12, 360, 75600, 174636000, 5244319080000, 2677277333530800000, 25968760179275365452000000, 5793445238736255798985527240000000, 37481813439427687898244906452608585200000000, 7517370874372838151564668004911177464757864076000000000, 55784440720968513813368002533861454979548176771615744085560000000000

Offset: 0

N. J. A. Sloane

Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).
Superprimorials, from primorials by analogy with superfactorials.
Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.
Subsequence of A130091. - Reinhard Zumkeller, May 06 2007
Hankel transform of A171448. - Paul Barry, Dec 09 2009
This might be a good place to explain the name "Chernoff sequence" since his name does not appear in the References or Links as of Mar 22 2014. - Jonathan Sondow, Mar 22 2014
Pickover (1992) named this sequence after Paul Chernoff of California, who contributed this sequence to his book. He was possibly referring to American mathematician Paul Robert Chernoff (1942 - 2017), a professor at the University of California. - Amiram Eldar, Jul 27 2020

			a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.

T. D. Noe, Table of n, a(n) for n=0..25

Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).
Cf. A002110, A051357.
A000142 counts divisors of superprimorials.
A000325 counts uniform divisors of superprimorials.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 is a sister-sequence.
A118914 has row a(n) equal to {1..n}.
A124010 has row a(n) equal to {n..1}.
A130091 lists numbers with distinct prime multiplicities.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
A336426 gives non-products of superprimorials.
Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.

a006939 n = a006939_list !! n
a006939_list = scanl1 (*) a002110_list -- Reinhard Zumkeller, Jul 21 2012

[1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // G. C. Greubel, Oct 14 2018

a := []; printlevel := -1; for k from 0 to 20 do a := [op(a),product(ithprime(i)^(k-i+1),i=1..k)] od; print(a);

Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* Harvey P. Dale, Jul 07 2011 *)
Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* Alonso del Arte, Sep 30 2011 *)

a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ Charles R Greathouse IV, Jul 25 2011

from math import prod
from sympy import prime
def A006939(n): return prod(prime(k)**(n-k+1) for k in range(1,n+1)) # Chai Wah Wu, Aug 12 2025

a(n) = m(1)*m(2)*m(3)*...*m(n), where m(n) = n-th primorial number. - N. J. A. Sloane, Feb 20 2005
a(0) = 1, a(n) = a(n - 1)p(n)#, where p(n)# is the n-th primorial A002110(n) (the product of the first n primes). - Alonso del Arte, Sep 30 2011
log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - Charles R Greathouse IV, Mar 14 2011
A181796(a(n)) = A000110(n+1). It would be interesting to have a bijective proof of this theorem, which is stated at A181796 without proof. See also A336420. - Gus Wiseman, Aug 03 2020

Corrected and extended by Labos Elemer, May 30 2001

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

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

The first term that is not a factorial number is a(180) = 12.
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.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

			The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

Andrew Howroyd, Table of n, a(n) for n = 1..4096
Wikipedia, Permutation pattern

Separations are counted by A003242 and ranked by A333489.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535 and ranked by A335448.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.

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

F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[,2])} \\ Andrew Howroyd, Feb 03 2021

A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, May 28 2021

nonn

Differs from A335448 in having a(x^2) = 0 and a(270) = 0.
These are permutations of the prime factors of n, counting multiplicity, with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins (x,x) is A345164, which is identical to this sequence except when n is the square of a prime.

			The permutations for n = 2, 6, 30, 180, 210, 300, 420, 720, 840:
  2   23   253   23253   2537   25253   23275   2323252   232527
      32   325   32325   2735   25352   25273   2325232   232725
           352   32523   3275   32525   25372   2523232   252327
           523   35232   3527   35252   27253             252723
                 52323   3725   52325   27352             272325
                         5273   52523   32527             272523
                         5372           32725             325272
                         5723           35272             327252
                         7253           37252             523272
                         7352           52327             527232
                                        52723             723252
                                        57232             725232
                                        72325
                                        72523
For example, there are no alternating permutations of the prime factors of 270 because the only anti-runs are {3,2,3,5,3} and {3,5,3,2,3}, neither of which is alternating, so a(270) = 0.

The version for permutations is A001250.
The extension to anti-run permutations is A335452.
The version for compositions is A344604.
The version for patterns is A344605.
Positions of zeros are A344653 (counted by A344654).
Not including twins (x,x) gives A345164.
A008480 counts permutations of prime indices (strict: A335489, rank: A333221).
A056239 adds up prime indices,  row sums of A112798.
A071321 and A071322 are signed sums of prime factors.
A316523 is a signed sum of prime multiplicities.
A316524 and A344616 are signed sums of prime indices.
A325534 counts separable partitions (ranked by A335433).
A325535 counts inseparable partitions (ranked by A335448).
A344740 counts partitions with an alternating permutation or twin (x,x).
Cf. A000041, A000607, A000961, A001222, A003242, A026424, A028260, A049774, A103919, A181796, A343938, A344652.

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,100}]

A261982 Number of compositions of n with some part repeated.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>Length[Split[#]]&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = A011782(n) - A032020(n).

G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020

cp's OEIS Frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A032020 Number of compositions (ordered partitions) of n into distinct parts.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI