# Euler–Catalan’s Number Triangle and Its Application

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Combinatorial Objects

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{n}\setminus \mathit{m}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | |||||||||

2 | 1 | 1 | ||||||||

3 | 2 | 2 | 1 | |||||||

4 | 5 | 5 | 3 | 1 | ||||||

5 | 14 | 14 | 9 | 4 | 1 | |||||

6 | 42 | 42 | 28 | 14 | 5 | 1 | ||||

7 | 132 | 132 | 90 | 48 | 20 | 6 | 1 | |||

8 | 429 | 429 | 297 | 165 | 75 | 27 | 7 | 1 | ||

9 | 1430 | 1430 | 1001 | 572 | 275 | 110 | 35 | 8 | 1 | |

10 | 4862 | 4862 | 3432 | 2002 | 1001 | 429 | 154 | 44 | 9 | 1 |

$\mathit{n}\setminus \mathit{m}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | |||||||||

1 | 1 | 0 | ||||||||

2 | 1 | 1 | 0 | |||||||

3 | 1 | 4 | 1 | 0 | ||||||

4 | 1 | 11 | 11 | 1 | 0 | |||||

5 | 1 | 26 | 66 | 26 | 1 | 0 | ||||

6 | 1 | 57 | 302 | 302 | 57 | 1 | 0 | |||

7 | 1 | 120 | 1191 | 2416 | 1191 | 120 | 1 | 0 | ||

8 | 1 | 247 | 4293 | 15,619 | 15,619 | 4293 | 247 | 1 | 0 | |

9 | 1 | 502 | 14,608 | 88,234 | 156,190 | 88,234 | 14,608 | 502 | 1 | 0 |

$\mathit{n}\setminus \mathit{m}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | |||||||||

1 | 1 | 1 | ||||||||

2 | 1 | 2 | 2 | |||||||

3 | 1 | 3 | 6 | 6 | ||||||

4 | 1 | 4 | 12 | 24 | 24 | |||||

5 | 1 | 5 | 20 | 60 | 120 | 120 | ||||

6 | 1 | 6 | 30 | 120 | 360 | 720 | 720 | |||

7 | 1 | 7 | 42 | 210 | 840 | 2520 | 5040 | 5040 | ||

8 | 1 | 8 | 56 | 336 | 1680 | 6720 | 20,160 | 40,320 | 40,320 | |

9 | 1 | 9 | 72 | 504 | 3024 | 15,120 | 60,480 | 181,440 | 362,880 | 362,880 |

$\mathit{n}\setminus \mathit{m}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

0 | 1 | ||||||||

1 | 1 | 0 | |||||||

2 | 3 | 1 | 0 | ||||||

3 | 19 | 10 | 1 | 0 | |||||

4 | 193 | 119 | 23 | 1 | 0 | ||||

5 | 2721 | 1806 | 466 | 46 | 1 | 0 | |||

6 | 49,171 | 34,017 | 10,262 | 1502 | 87 | 1 | 0 | ||

7 | 1,084,483 | 770,274 | 255,795 | 47,020 | 4425 | 162 | 1 | 0 | |

8 | 28,245,729 | 20,429,551 | 7,235,853 | 1,539,939 | 193,699 | 12,525 | 303 | 1 | 0 |

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**MDPI and ACS Style**

Shablya, Y.; Kruchinin, D.
Euler–Catalan’s Number Triangle and Its Application. *Symmetry* **2020**, *12*, 600.
https://doi.org/10.3390/sym12040600

**AMA Style**

Shablya Y, Kruchinin D.
Euler–Catalan’s Number Triangle and Its Application. *Symmetry*. 2020; 12(4):600.
https://doi.org/10.3390/sym12040600

**Chicago/Turabian Style**

Shablya, Yuriy, and Dmitry Kruchinin.
2020. "Euler–Catalan’s Number Triangle and Its Application" *Symmetry* 12, no. 4: 600.
https://doi.org/10.3390/sym12040600