A249543 -id:A249543 - cp's OEIS frontend

A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 20, 90, 24, 1, 1, 1, 70, 1680, 2520, 120, 1, 1, 1, 252, 34650, 369600, 113400, 720, 1, 1, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1

Offset: 0

Robert G. Wilson v, Jan 05 2013

From Tilman Piesk, Oct 28 2014: (Start)
Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.
In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:
  m\n:0   1       2            3                4                    5
  0:  1   1       1            1                1                    1 ... A000012;
  1:  1   1       1            1                1                    1 ... A000012;
  2:  1   2       6           20               70                  252 ... A000984;
  3:  1   6      90         1680            34650               756756 ... A006480;
  4:  1  24    2520       369600         63063000          11732745024 ... A008977;
  5:  1 120  113400    168168000     305540235000      623360743125120 ... A008978;
  6:  1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;
with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).
A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).
(End)

			T(3,5) = (3*5)!/(5!^3) = 756756 = A014609(3) = A006480(5) is the number of permutations of a multiset that contains 3 different elements 5 times, e.g., {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3}.

Tilman Piesk, First 54 rows of the triangle, flattened
T. Chappell, A. Lascoux, S. Ole Warnaar, and W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.
Tilman Piesk, Array for indices 0..16
Tilman Piesk, PHP code used to create the b-file
Tilman Piesk, Illustration of the multisets for m,n=0..4
Wikipedia, Permutations of multisets, Pascal matrix and simplex

Cf. A089759 (transposed), A141906 (subtriangle), A120666 (subtriangle transposed), A060538 (1st row/column removed).
Rows: A000012, A000984, A006480, A006480, A008978, A008979, A071549, A071550, A071551, A071552.
Columns: A000012, A000142, A000680, A014606, A014608, A014609, A248814, A172603, A172609, A172613.
Main diagonal gives: A034841.
Row sums of the triangle: A248827.

[Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 26 2022

T[n_, k_]:= (k*n)!/(n!)^k; Table[T[n, k-n], {k, 9}, {n, 0, k-1}]//Flatten

def A187783(n,k): return gamma(k*(n-k)+1)/(factorial(n-k))^k
flatten([[A187783(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Dec 26 2022

T(m,n) = (m*n)!/(n!)^m.

A060540(m,n) = T(m,n)/m! . - R. J. Mathar, Jun 21 2023

Row m=0 prepended by Tilman Piesk, Oct 28 2014

A194602 Integer partitions interpreted as binary numbers.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 21, 23, 27, 31, 43, 47, 55, 63, 85, 87, 91, 95, 111, 119, 127, 171, 175, 183, 191, 219, 223, 239, 255, 341, 343, 347, 351, 367, 375, 383, 439, 447, 479, 495, 511, 683, 687, 695, 703, 731, 735, 751, 767, 879, 887, 895, 959, 991, 1023, 1365, 1367, 1371, 1375, 1391

Offset: 0

Tilman Piesk, Aug 30 2011

The 2^(n-1) compositions of n correspond to binary numbers, and the partitions of n can be seen as compositions with addends ordered by size, so they also correspond to binary numbers.
The finite sequence for partitions of n (ordered by size) is the beginning of the sequence for partitions of n+1, which leads to an infinite sequence.
From Tilman Piesk, Jan 30 2016: (Start)
It makes sense to regard the positive values as a triangle with row lengths A002865(n) and row numbers n>=2. In this triangle row n contains all partitions of n with non-one addends only. See link "Triangle with Young diagrams".
This sequence contains all binary palindromes with m runs of n ones separated by single zeros. They are ordered in the array A249544. All the rows and columns of this array are subsequences of this sequence, notably its top row (A000225, the powers of two minus one).
Sequences by Omar E. Pol: The "triangle" A210941 defines the same sequence of partitions. Its n-th row shows the non-one addends of the n-th partition. There are A194548(n) of them, and A141285(n) is the largest among them. (The "triangle" A210941 does not actually form a triangle, but A210941 and A141285 do.) Note that the offset of these sequences is 1 and not 0.
(End)
Numbers whose binary representation has runs of '1's of weakly increasing length (with trailing '0's (introducing a run of length 0) forbidden, i.e., only odd terms beyond 0). - M. F. Hasler, May 14 2020

			From _Joerg Arndt_, Nov 17 2012: (Start)
With leading zeros included, the first A000041(n) terms correspond to the list of partitions of n as nondecreasing compositions in lexicographic order.
For example, the first A000041(10)=42 terms correspond to the partitions of 10 as follows (dots for zeros in the binary expansions):
[ n]   binary(a(n))  a(n)  partition
[ 0]   ..........     0    [ 1 1 1 1 1 1 1 1 1 1 ]
[ 1]   .........1     1    [ 1 1 1 1 1 1 1 1 2 ]
[ 2]   ........11     3    [ 1 1 1 1 1 1 1 3 ]
[ 3]   .......1.1     5    [ 1 1 1 1 1 1 2 2 ]
[ 4]   .......111     7    [ 1 1 1 1 1 1 4 ]
[ 5]   ......1.11    11    [ 1 1 1 1 1 2 3 ]
[ 6]   ......1111    15    [ 1 1 1 1 1 5 ]
[ 7]   .....1.1.1    21    [ 1 1 1 1 2 2 2 ]
[ 8]   .....1.111    23    [ 1 1 1 1 2 4 ]
[ 9]   .....11.11    27    [ 1 1 1 1 3 3 ]
[10]   .....11111    31    [ 1 1 1 1 6 ]
[11]   ....1.1.11    43    [ 1 1 1 2 2 3 ]
[12]   ....1.1111    47    [ 1 1 1 2 5 ]
[13]   ....11.111    55    [ 1 1 1 3 4 ]
[14]   ....111111    63    [ 1 1 1 7 ]
[15]   ...1.1.1.1    85    [ 1 1 2 2 2 2 ]
[16]   ...1.1.111    87    [ 1 1 2 2 4 ]
[17]   ...1.11.11    91    [ 1 1 2 3 3 ]
[18]   ...1.11111    95    [ 1 1 2 6 ]
[19]   ...11.1111   111    [ 1 1 3 5 ]
[20]   ...111.111   119    [ 1 1 4 4 ]
[21]   ...1111111   127    [ 1 1 8 ]
[22]   ..1.1.1.11   171    [ 1 2 2 2 3 ]
[23]   ..1.1.1111   175    [ 1 2 2 5 ]
[24]   ..1.11.111   183    [ 1 2 3 4 ]
[25]   ..1.111111   191    [ 1 2 7 ]
[26]   ..11.11.11   219    [ 1 3 3 3 ]
[27]   ..11.11111   223    [ 1 3 6 ]
[28]   ..111.1111   239    [ 1 4 5 ]
[29]   ..11111111   255    [ 1 9 ]
[30]   .1.1.1.1.1   341    [ 2 2 2 2 2 ]
[31]   .1.1.1.111   343    [ 2 2 2 4 ]
[32]   .1.1.11.11   347    [ 2 2 3 3 ]
[33]   .1.1.11111   351    [ 2 2 6 ]
[34]   .1.11.1111   367    [ 2 3 5 ]
[35]   .1.111.111   375    [ 2 4 4 ]
[36]   .1.1111111   383    [ 2 8 ]
[37]   .11.11.111   439    [ 3 3 4 ]
[38]   .11.111111   447    [ 3 7 ]
[39]   .111.11111   479    [ 4 6 ]
[40]   .1111.1111   495    [ 5 5 ]
[41]   .111111111   511    [ 10 ]
(End)

Tilman Piesk, Table of n, a(n) for n = 0..8348
Tilman Piesk, Same table with binary strings and non-one addends
Tilman Piesk, Triangle with Young diagrams (n = 2..20).
Tilman Piesk, Integer partitions and Permutations and partitions in the OEIS
Tilman Piesk, Python functions, keynum_to_valnum(n) = a(n), valnum_to_keynum(a(n)) = n.
Li-yao Xia, Identities for A194602
Index entries for sequences related to binary expansion of n

Cf. A000041 (partition numbers).
Cf. A002865 (row lengths).
Cf. A002450, A000225 (subsequences).
Cf. A249544 (rows and columns are subsequences).
Cf. A210941, A194548, A141285, A228354, A326956.

lim = 12;
Sort[FromDigits[Reverse@ #, 2] & /@
   Map[If[Length@ # == 0, {0}, Flatten@ Most@ #] &@
     Riffle[#, Table[0, Length@ #]] &,
     Map[Table[1, # - 1] &,
       Sort@ IntegerPartitions@ lim /. 1 -> Nothing, {2}]]]
(* Michael De Vlieger, Feb 14 2016 *)

isA194602(n) = if(!n,1,if(!(n%2),0,my(prl=0,rl=0); while(n, if(0==(n%2),if((prl && rl>prl)||0==(n%4), return(0)); prl=rl; rl=0, rl++); n >>= 1); ((0==prl)||(rl<=prl)))); \\ - Antti Karttunen, Dec 06 2021

a( A000041(n)-1 ) = A000225(n-1) for n>=1. - Tilman Piesk, Apr 16 2012
a( A000041(2n-1) ) = A002450(n)  for n>=1. - Tilman Piesk, Apr 16 2012
a( A249543 ) = A249544. - Tilman Piesk, Oct 31 2014
a(n) = A228354(1+n) - 1. - Antti Karttunen, Dec 06 2021

Comments edited by Li-yao Xia, May 13 2014

Incorrect PARI-program removed by Antti Karttunen, Dec 09 2021

A249544 Array read by antidiagonals: T(m,n) read in binary is a palindrome with m runs of n ones separated by single zeros.

Original entry on oeis.org

1, 3, 5, 7, 27, 21, 15, 119, 219, 85, 31, 495, 1911, 1755, 341, 63, 2015, 15855, 30583, 14043, 1365, 127, 8127, 128991, 507375, 489335, 112347, 5461, 255, 32639, 1040319, 8255455, 16236015, 7829367, 898779, 21845, 511, 130815, 8355711

Offset: 1

Tilman Piesk, Oct 31 2014

The entries in this array are all in A194602, and therefore can be interpreted as integer partitions: T(m,n) is the integer partition with m times the addend n+1, and no other non-one addends. The array A249543 contains the corresponding indices of A194602.

			Array starts:                                          Binary:
  n    1      2       3         4          5
m
1      1      3       7        15         31               1        11          111
2      5     27     119       495       2015             101     11011      1110111
3     21    219    1911     15855     128991           10101  11011011  11101110111
4     85   1755   30583    507375    8255455
5    341  14043  489335  16236015  528349151

Tilman Piesk, First 113 rows of the triangle, flattened

Cf. A249543, A194602; Rows: A000225, A129868; Columns: A002450, A083713.

A249544($m, $n) {
// a                  b            c
// ( 2^(n+1)^m -1 ) * ( 2^n -1 ) / ( 2^(n+1) -1 )
$a = gmp_sub( gmp_pow( gmp_pow(2,$n+1), $m ), 1 );
$b = gmp_sub( gmp_pow(2,$n), 1 );
$c = gmp_sub( gmp_pow(2,$n+1), 1 );
$return = gmp_div_q( gmp_mul($a,$b), $c );
return gmp_strval($return);
}

T(m,n) = ( 2^(n+1)^m -1 ) * ( 2^n -1 ) / ( 2^(n+1) -1 ).

cp's OEIS Frontend

A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A194602 Integer partitions interpreted as binary numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A249544 Array read by antidiagonals: T(m,n) read in binary is a palindrome with m runs of n ones separated by single zeros.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PHP

Formula