A008302 -id:A008302 - cp's OEIS frontend

A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).

1, 1, 2, 6, 23, 1, 103, 11, 4, 2, 513, 88, 56, 32, 14, 7, 9, 0, 0, 1, 2761, 638, 543, 341, 235, 138, 173, 51, 42, 47, 34, 6, 17, 4, 0, 7, 1, 0, 2, 15767, 4478, 4600, 3119, 2658, 1710, 2180, 972, 975, 877, 771, 356, 542, 233, 184, 266, 157, 81, 130, 41, 60, 49, 16, 16, 37, 8, 9, 13, 3, 0, 10, 1, 0, 0, 0, 0, 1

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the patterns 2134, 3421, and 4312.

First column is A005802.

			Table begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      11,       4,       2;
   6 |     513,      88,      56,      32,      14,       7,       9, ...
   7 |    2761,     638,     543,     341,     235,     138,     173, ...
   8 |   15767,    4478,    4600,    3119,    2658,    1710,    2180, ...
   9 |   94359,   31199,   36691,   26602,   25756,   17628,   22984, ...
  10 |  586590,  218033,  284370,  218957,  231390,  166338,  221429, ...
  11 | 3763290, 1535207, 2174352, 1767837, 1994176, 1496134, 2028316, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A005802, A028723.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342864 (1432), A342865 (1234).

A342864 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1432. 0 <= k <= A100354(n).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 103, 11, 5, 0, 1, 513, 87, 68, 17, 18, 10, 0, 4, 2, 0, 1, 2761, 625, 626, 268, 274, 138, 112, 58, 51, 44, 31, 9, 15, 8, 12, 0, 5, 0, 0, 0, 3, 15767, 4378, 5038, 2781, 3060, 1697, 1817, 1036, 964, 773, 656, 450, 379, 320, 285, 148, 237, 97, 98, 55, 68, 61, 23, 30, 30, 13, 30, 0, 0, 0, 16, 0, 10, 0, 0, 1, 0, 0, 0, 0, 2

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the patterns 2341, 3214, and 4123.

First column is A005802.

			Table begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      11,       5,       0,       1;
   6 |     513,      87,      68,      17,      18,      10,       0, ...
   7 |    2761,     625,     626,     268,     274,     138,     112, ...
   8 |   15767,    4378,    5038,    2781,    3060,    1697,    1817, ...
   9 |   94359,   30671,   38541,   24731,   28881,   17943,   21193, ...
  10 |  586590,  216883,  289785,  205853,  251051,  170941,  211942, ...
  11 | 3763290, 1552588, 2172387, 1663964, 2096207, 1535129, 1954751, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A005802, A100354.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342863 (1243), A342865 (1234).

A342865 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1234. 0 <= k <= A000332(n).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 103, 12, 4, 0, 0, 1, 513, 102, 63, 10, 6, 12, 8, 0, 0, 5, 0, 0, 0, 0, 0, 1, 2761, 770, 665, 196, 146, 116, 142, 46, 10, 72, 32, 24, 0, 13, 0, 12, 18, 0, 0, 10, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the pattern 4321.

First column is A005802.

			Table begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      12,       4,       0,       0,       1;
   6 |     513,     102,      63,      10,       6,      12,       8, ...
   7 |    2761,     770,     665,     196,     146,     116,     142, ...
   8 |   15767,    5545,    5982,    2477,    2148,    1204,    1782, ...
   9 |   94359,   39220,   49748,   25886,   25190,   13188,   19936, ...
  10 |  586590,  276144,  396642,  244233,  260505,  142550,  210663, ...
  11 | 3763290, 1948212, 3089010, 2167834, 2493489, 1476655, 2136586, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A000332, A005802.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342863 (1243), A342864 (1432).

A381529 T(n,k) is the number of permutations of [n] having exactly k pairs of integers i=0, 0<=k<=A125811(n)-1, read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 15, 5, 4, 54, 21, 24, 16, 5, 235, 89, 118, 112, 101, 35, 28, 2, 1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14, 7790, 2106, 3023, 3529, 4410, 4463, 4600, 3012, 2789, 1933, 1438, 629, 442, 122, 34, 57581, 12529, 17693, 20980, 27208, 30064, 35359, 33332, 28137, 24970, 22850, 17148, 14272, 8645, 5639, 3684, 1809, 664, 282, 34

Offset: 0

Alois P. Heinz, Feb 26 2025

			T(4,0) = 15: (1)(2)(3)(4), (1,2)(3)(4), (1)(2,3)(4), (1)(2)(3,4), (1,2)(3,4), (1,2,3)(4), (1,3,2)(4), (1)(2,3,4), (1)(2,4,3), (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2).
T(4,1) = 5: (1)(2,4)(3), (1,2,4)(3), (1,4,2)(3), (1,3)(2)(4), (1,3)(2,4).
T(4,2) = 4: (1,4)(2)(3), (1,4)(2,3), (1,3,4)(2), (1,4,3)(2).
Triangle T(n,k) begins:
     1;
     1;
     2;
     5,   1;
    15,   5,   4;
    54,  21,  24,  16,   5;
   235,  89, 118, 112, 101,  35,  28,   2;
  1237, 408, 577, 633, 719, 585, 402, 239, 167, 59, 14;
  ...

Alois P. Heinz, Rows n = 0..55, flattened
Wikipedia, Inversion
Wikipedia, Permutation

Columns k=0-1 give: A051295, A381539.
Row sums give A000142.
Row lengths give A125811.
Last elements of rows give A381531.
Main diagonal gives A381545.
Cf. A008302, A125810 (similar for set partitions), A126673, A381299 (similar for ordered set partitions).

b:= proc(o, u, t) option remember; expand(`if`(u+o=0, max(0, t-1)!,
     `if`(t>0, b(u+o, 0$2)*(t-1)!, 0)+add(x^(u+j-1)*
        b(o-j, u+j-1, t+1), j=`if`(t=0, 1, 1..o))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..10);

Sum_{k>=1} k * T(n,k) = A126673(n)/2.

A005283 Number of permutations of (1,...,n) having n-5 inversions (n>=5).

Original entry on oeis.org

1, 5, 20, 76, 285, 1068, 4015, 15159, 57486, 218895, 836604, 3208036, 12337630, 47572239, 183856635, 712033264, 2762629983, 10736569602, 41788665040, 162869776650, 635562468075, 2482933033659, 9710010151831, 38008957336974, 148912655255315, 583885852950802

Offset: 5

N. J. A. Sloane

nonn

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

			a(6)=5 because we have 213456, 132456, 124356, 123546, and 123465.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
R. K. Guy, personal communication.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 5..1000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.

Cf. A008302, A000707, A001892, A001893, A001894, A005284, A005285, A048651.

f := (x,n)->product((1-x^j)/(1-x),j=1..n); seq(coeff(series(f(x,n),x,n+2),x,n-5), n=5..40); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-5}],{n,5,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-6)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

A005284 Number of permutations of (1,...,n) having n-6 inversions (n>=6).

Original entry on oeis.org

1, 6, 27, 111, 440, 1717, 6655, 25728, 99412, 384320, 1487262, 5762643, 22357907, 86859412, 337879565, 1315952428, 5131231668, 20029728894, 78265410550, 306109412100, 1198306570554, 4694809541046, 18407850118383

Offset: 6

N. J. A. Sloane

nonn

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

			a(7)=6 because we have 2134567, 1324567, 1243567, 1235467, 1234657 and 1234576.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
R. K. Guy, personal communication.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 6..1000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.

Cf. A008302, A000707, A001892, A001893, A001894, A005283, A005285, A048651.

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; seq(g(j+6,j),j=0..30); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-6}],{n,6,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-7)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

A005285 Number of permutations of (1,...,n) having n-7 inversions (n>=7).

Original entry on oeis.org

1, 7, 35, 155, 649, 2640, 10569, 41926, 165425, 650658, 2554607, 10020277, 39287173, 154022930, 603919164, 2368601685, 9293159292, 36476745510, 143239635450, 562744102479, 2211876507387, 8697839966552, 34218338900591

Offset: 7

N. J. A. Sloane

nonn

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

			a(8)=7 because we have 21345678, 13245678, 12435678, 12354678, 12346578, 12345768, and 12345687.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
R. K. Guy, personal communication.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 7..1000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.

Cf. A008302, A000707, A001892, A001893, A001894, A005283, A005284, A048651.

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; seq(g(j+7,j),j=0..30); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-7}],{n,7,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-8)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0

Offset: 0

Gus Wiseman, Aug 03 2020

Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  0
  1  2  1  2  1
  1  3  1  3  2  0
  1  3  2  5  3  3  2  1
  1  4  2  7  4  4  3  2  0
  1  4  2  7  4  5  7  7  6  3  2  0
  1  4  2  8  8  9 10 11 11  7  8  5  2  0
  1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
  1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
  1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
Row n = 7 counts the following divisors:
  1  2  4  8   16  48   144  720   {}
     3  9  12  24  72   360  1008
     5     18  40  80   504
     7     20  56  112
           28
           45
           63

A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

A005288 a(n) = C(n,5) + C(n,4) - C(n,3) + 1, n >= 7.

Original entry on oeis.org

3, 22, 71, 169, 343, 628, 1068, 1717, 2640, 3914, 5629, 7889, 10813, 14536, 19210, 25005, 32110, 40734, 51107, 63481, 78131, 95356, 115480, 138853, 165852, 196882, 232377, 272801, 318649, 370448, 428758, 494173, 567322

Offset: 6

N. J. A. Sloane

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 15.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241-242. (Annotated scanned copy)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008302.

Join[{3},Table[Binomial[n,5]+Binomial[n,4]-Binomial[n,3]+1,{n,7,50}]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3,22,71,169,343,628,1068},50] (* Harvey P. Dale, Aug 30 2021 *)

a(n) = C(n+3, 5) - C(n+2, 3) + C(n, 0).
G.f.: 3*x^6 -x^7*(x-2)*(2*x^4-11*x^3+24*x^2-25*x+11)/(x-1)^6. Simon Plouffe in his 1992 dissertation
a(n) = (n+4)*(n-3)*(n^3-6*n^2+3*n-10)/120, n >= 7. - R. J. Mathar, May 19 2013

A336865 Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2020

Row lengths are A073093(n) = A001222(n) + 1.

			The triangle begins as follows. The n-th row is shown to the right of "n:".
     1: (1)          16: (1,1,1,1,1)    31: (1,1)
     2: (1,1)        17: (1,1)          32: (1,1,1,1,1,1)
     3: (1,1)        18: (1,2,1,1)      33: (1,2,0)
     4: (1,1,1)      19: (1,1)          34: (1,2,0)
     5: (1,1)        20: (1,2,1,1)      35: (1,2,0)
     6: (1,2,0)      21: (1,2,0)        36: (1,2,2,2,0)
     7: (1,1)        22: (1,2,0)        37: (1,1)
     8: (1,1,1,1)    23: (1,1)          38: (1,2,0)
     9: (1,1,1)      24: (1,2,1,2,1)    39: (1,2,0)
    10: (1,2,0)      25: (1,1,1)        40: (1,2,1,2,1)
    11: (1,1)        26: (1,2,0)        41: (1,1)
    12: (1,2,1,1)    27: (1,1,1,1)      42: (1,3,0,0)
    13: (1,1)        28: (1,2,1,1)      43: (1,1)
    14: (1,2,0)      29: (1,1)          44: (1,2,1,1)
    15: (1,2,0)      30: (1,3,0,0)      45: (1,2,1,1)
Row n = 72 counts the following divisors:
  1  2  4   8  24  72
     3  9  12
           18
Row n = 1200 counts the following divisors:
  1  2   4   8  16   48  400  1200
     3  25  12  24   80  600
     5      20  40  200
            50
            75

A073093 gives row lengths.
A130092 gives positions of rows ending with 0.
A146291 is the version not requiring distinct prime multiplicities.
A181796 gives row sums.
A336499 is the restriction to factorial numbers.
A001222 counts prime factors, counting multiplicity.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336423 counts chains using A130091.
Cf. A000005, A001221, A098859, A118914, A124010, A336422, A336500, A336571.

Table[Length[Select[Divisors[n],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,20},{k,0,PrimeOmega[n]}]

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A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).

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A342864 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1432. 0 <= k <= A100354(n).

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A342865 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1234. 0 <= k <= A000332(n).

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A381529 T(n,k) is the number of permutations of [n] having exactly k pairs of integers i=0, 0<=k<=A125811(n)-1, read by rows.

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A005283 Number of permutations of (1,...,n) having n-5 inversions (n>=5).

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A005284 Number of permutations of (1,...,n) having n-6 inversions (n>=6).

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A005285 Number of permutations of (1,...,n) having n-7 inversions (n>=7).

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A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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