cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A346189 a(n) is the number of permutations on [n] with no strong fixed points or small descents.

Original entry on oeis.org

0, 0, 2, 6, 34, 214, 1550, 12730, 116874, 1187022, 13219550, 160233258, 2100360778, 29610224590, 446789311934, 7185155686666, 122690711149290, 2217055354281582, 42269657477711198, 847998698508705834, 17857221256001240458, 393839277313540073230, 9078806210245773668990, 218340709713567352161226
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n = 4, the a(4) = 6 permutations on [4] with no strong fixed points or small descents: {(2,3,4,1),(3,4,1,2),(4,1,2,3),(3,1,4,2),(2,4,1,3),(4,2,3,1)}.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346198, A346199, A346204.

Programs

Formula

For n > 3, a(n) = b(n) - b(n-1) - Sum{i=4..n}(a(i-1)*b(n-i)) where b(n) = A000255(n-1) and b(0) = 1.

A346198 a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.

Original entry on oeis.org

0, 1, 1, 8, 43, 283, 2126, 17947, 168461, 1741824, 19684171, 241506539, 3198239994, 45482655683, 691471698917, 11193266251700, 192238116358427, 3491633681792507, 66875708261486766, 1347168876070616179, 28474546456352896021, 630130731702950549248, 14570725407559756078387, 351411668456841530417027
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n = 4, the a(4) = 8 permutations on [4] with no strong fixed points but has small descents: {([2, 1], [4, 3]), (2, [4, 3], 1), ([3, 2], 4, 1), (3, 4, [2, 1]), (4, 1, [3, 2]), (4, [2, 1], 3), ([4, 3], 1, 2), (<4, 3, 2, 1>)} []small descent, <>consecutive small descents.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346199, A346204.

Programs

Formula

For n > 2, a(n) = b(n)-c(n) where b(n) = A052186(n-1), c(n) = A346189(n).

A346199 a(n) is the number of permutations on [n] with at least one strong fixed point and no small descents.

Original entry on oeis.org

1, 1, 1, 5, 19, 95, 569, 3957, 31455, 281435, 2799981, 30666153, 366646995, 4751669391, 66348304849, 992975080813, 15856445382119, 269096399032035, 4836375742967861, 91766664243841393, 1833100630242606203, 38452789552631651191, 845116020421125048153
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n = 4, the a(4) = 5 permutations on [4] with strong fixed points but no small descents: {(1*, 2*, 3*, 4*), (1*, 3, 4, 2), (1*, 4, 2, 3), (2, 3, 1, 4*), (3, 1, 2, 4*)} where * marks strong fixed points.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, A346204.

Programs

Formula

a(n) = b(n-1) + Sum_{i=4..n} A346189(i-1)*b(n-i) where b(n) = A000255(n).
Showing 1-3 of 3 results.

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