login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A218577 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with maximal element k-1. 7
1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 11, 1, 1, 31, 90, 74, 20, 1, 1, 63, 301, 402, 209, 37, 1, 1, 127, 966, 1951, 1629, 590, 70, 1, 1, 255, 3025, 8869, 10839, 6430, 1685, 135, 1, 1, 511, 9330, 38720, 65720, 56878, 25313, 4870, 264, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums are A022493.

Second column is A000225 (2^n - 1).

Third column appears to be A000392 (Stirling numbers S(n,3)).

Second diagonal (from the right) appears to be A006127 (2^n + n).

REFERENCES

William Y. C. Chen, Alvin Y.L. Dai, Theodore Dokos, Tim Dwyer and Bruce E. Sagan, On 021-Avoiding Ascent Sequences, The Electronic Journal of Combinatorics. Volume 20, Issue 1 (2013), #P76.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened (first 15 rows from Joerg Arndt)

Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, Sergey Kitaev, (2+2)-free posets, ascent sequences and pattern avoiding permutations, arXiv:0806.0666 [math.CO]

EXAMPLE

Triangle starts:

1;

1,    1;

1,    3,     1;

1,    7,     6,      1;

1,   15,    25,     11,      1;

1,   31,    90,     74,     20,      1;

1,   63,   301,    402,    209,     37,      1;

1,  127,   966,   1951,   1629,    590,     70,     1;

1,  255,  3025,   8869,  10839,   6430,   1685,   135,     1;

1,  511,  9330,  38720,  65720,  56878,  25313,  4870,   264,   1;

1, 1023, 28501, 164676, 376114, 444337, 292695, 99996, 14209, 521, 1;

...

The 53 ascent sequences of length 5 are (dots for zeros):

[ #]     ascent-seq.   #max digit

[ 1]    [ . . . . . ]   0

[ 2]    [ . . . . 1 ]   1

[ 3]    [ . . . 1 . ]   1

[ 4]    [ . . . 1 1 ]   1

[ 5]    [ . . . 1 2 ]   2

[ 6]    [ . . 1 . . ]   1

[ 7]    [ . . 1 . 1 ]   1

[ 8]    [ . . 1 . 2 ]   2

[ 9]    [ . . 1 1 . ]   1

[10]    [ . . 1 1 1 ]   1

[11]    [ . . 1 1 2 ]   2

[12]    [ . . 1 2 . ]   2

[13]    [ . . 1 2 1 ]   2

[14]    [ . . 1 2 2 ]   2

[15]    [ . . 1 2 3 ]   3

[16]    [ . 1 . . . ]   1

[17]    [ . 1 . . 1 ]   1

[18]    [ . 1 . . 2 ]   2

[19]    [ . 1 . 1 . ]   1

[20]    [ . 1 . 1 1 ]   1

[21]    [ . 1 . 1 2 ]   2

[22]    [ . 1 . 1 3 ]   3

[23]    [ . 1 . 2 . ]   2

[24]    [ . 1 . 2 1 ]   2

[25]    [ . 1 . 2 2 ]   2

[26]    [ . 1 . 2 3 ]   3

[27]    [ . 1 1 . . ]   1

[28]    [ . 1 1 . 1 ]   1

[29]    [ . 1 1 . 2 ]   2

[...]

[49]    [ . 1 2 3 . ]   3

[50]    [ . 1 2 3 1 ]   3

[51]    [ . 1 2 3 2 ]   3

[52]    [ . 1 2 3 3 ]   3

[53]    [ . 1 2 3 4 ]   4

There is 1 sequence with maximum zero, 15 with maximum one, etc.,

therefore the fifth row is 1, 15, 25, 11, 1.

CROSSREFS

Cf. A022493 (number of ascent sequences), A137251 (ascent sequences with k ascents), A175579 (ascent sequences with k zeros).

Cf. A218579 (ascent sequences with last zero at position k-1), A218580 (ascent sequences with first occurrence of the maximal value at position k-1), A218581 (ascent sequences with last occurrence of the maximal value at position k-1).

Sequence in context: A154959 A080417 A008277 * A193387 A185982 A263858

Adjacent sequences:  A218574 A218575 A218576 * A218578 A218579 A218580

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Nov 03 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.